Love to welcome everybody back to the independent investor channel for a weekly update here on Hylian. This is going to be very important to earmark up to this point where progress has us looking at the beginning of 2023. I think we are in, well, we are in quiet period right now before the earnings approach. My expectations for this earnings call are muted. I don't think it's of paramount importance at this point in what I would consider to be a transformational phase going back from 2022 to where we are now fighting through the first half of 2023 and marching into an inevitable first look in the back half of 2023 and moving into Q1 2024. I think in retrospect, anybody who has put in the time on this project up to this point, I really want to try to get ahead of some expectations. Let me just put aside a few things here just so we can speak on the lines of total potential outcomes for this company. Okay. As of late, the stock has shown some resiliency. As a matter of fact, some upside momentum. I think there's been some renewed confidence in the market coming into 2023. Whether or not that lasts, whether or not the Fed actually does follow through with their final two rate hikes based on the metrics that they are looking at with the employment status in this country is yet to be seen. I don't invest along those lines. I do think there's some element of anticipation that those will dissipate within the next coming months if we were going to forecast 12 to 18 months into the future, which is where the stock market lives. Unfortunately, a lot of people want to live in the present and they want to try to make sense of day-to-day fluctuation in markets and it's a futile activity. All too often, the investor needs to actually take a step back and identify what it takes to actually be an investor. Being an investor in this company is exactly what is going to carry you through in whatever goals or personal thresholds of wealth that you're looking to obtain. Could the stock go down and out of business and highly unceased to exist? Sure. Let's just get that right out of the way. Could it continue to remain static, having proven over the last three years that this company has an inability to actually appreciate in stock price? Sure. Let's just put that out of the way for a second. Then we can come on every week and we can mull around a stock that has no hope. It has everything going against it. It has a story to it, yes, but that story is broken in so far as the world cannot find a fit for highly unholdings anywhere in the Class 8 space. Why? I have no idea, but we want to be holistic with our application here and discuss the real potential and remind people that this is in fact stock market investing at its core. What you think is probably going to happen in the stock market probably will not happen. That has played true of the last three years with this company. There's no way that you could have ever thought that if you were bullish on the company that the stock price would sink to these levels. Now, there's people out there that would say, you're wrong. I called it from the beginning. No, you didn't. You guessed. If you feel good about looking in the mirror and convincing yourself that you guessed right, no problem. The stock market doesn't reward past behavior. It only rewards the what if. The where a company is going to potentially go. From day to day, the stock market is continually looking to assess a fair value on a company that came to public markets through the SPAC process using funding to get them where they are. Now, you can go back and forth and criticize. You can be satisfied with where they are currently at their current juncture. I have a lot on both sides of the house critical in some cases, and also very, very satisfied in looking at this company really has been null and void of any controversy. They've done what they've needed to do. Some might suggest slower than what they projected. Yes, but we are where we are. You have the ability as a share owner or a follower of the company to sit back, offer your constructive criticism as to where you think that it should be now, which is kind of futile because the company has done what it's done up to this point, and there you go. Here we sit staring down an eventual what I consider to be a first look going into the back half of 2023. I can convey a little bit of my sentiment feeling right now at the beginning of 2023. I'm not going to lie to you. I feel pretty good. I feel pretty good. I think if there's a potential to have the worst behind us, the coming months and the coming year will give us that answer. We will be in one of three places at the end of this year. We will be licking our chops because we're licking our wounds because the stock again has ceased to exist. The stock has, or the company has failed to follow through on many of its promises going forward in its ability to be a leader in electrified powertrain solutions. That's their goal. So if they fail on that, company will cease to exist. Or by nature of the progress and the rollout, etc., that we all expect in some form or fashion to materialize in 2023 does not come to fruition. However, there's been marked progress along the line here in 2023. Enough to keep the stock afloat or maybe get a little bit of appreciation or depreciation. Again, I don't put a lot of merit in between this range that's been defined between $2, 5, 6, $10 even. If we can finish above 2023, above $10, I would consider that a massive, massive win. Is that going to happen? That represents a massive increase from these levels? I really don't know. I want to put this video out today and get people emotionally and technically conditioned with their perspective to have fair expectations about how quickly this company can move. And dare I suggest that perhaps maybe what we're all expecting for an inevitable end of 2023 and beginning of 2024 probably stands the most chance of disappointing investors at this particular juncture than actually being this enormous catalyst that I think all investors in the company would agree would be nice. But in all reality, it's been very, very evident to me that what they are trying to do is going to take time. If you don't have time to be an investor in this company, you could certainly segue your dollars into another opportunity. Good luck in trying to find the opportunity because I'm not going to rule out the potential that this thing shoots up 100, 200% within a week. I've seen it happen many, many times in stock market investing, anything is possible. So we want to be mentally prepared and have some discussion before those presumed outcomes actually happen so we can sit calmly on a Sunday and discuss what it is in preparation of those potential outcomes, how you're going to react to your specific holding. Very, very important for me with the body of work that I've put in on this project to understand that I think the majority of the wealth is going to be made on this company ultra long. I think retail investors often suffer from a disease of immediate satisfaction. Now you can get immediate satisfaction and a lot of things in life go into the bedroom and lock the door for an hour, immediate satisfaction. I think unfortunately for a lot of people I think people assimilate and they take what has worked for them in identifying a problem and going and getting the appropriate tool and tinkering with the problem or hitting it on the head with a hammer and fixing the problem or something breaks and we automatically go into emergency mode to fix whatever it is is broken, our appliances, our bodies, our cars, whatever that is. Stock market investing is like the most inhuman activity that we can possibly engage in. And I really want people to understand that if this company ends up increasing from here a hundred percent goes to six dollars. What is it that you're going to do with your stock? Now you're free to do whatever you want. The cool thing about people when they tune it to me, they understand I'm an independent mind and I'm not influenced by moves in the stock market. I have a goal for where I think that this company could be worth on the conservative side and I don't think they need to do a whole lot. Now the degree by which they move from almost zero in sales now to where I think they need to go is going to define the next 10 years of the company. I think it is the most critical. I think the most risk is on the table right now. I think when we're evaluating the for certain that I have for you guys in this video and the prove it moments or the for speculation category with highly on holdings, I think you could come up with a pretty good laundry list on either side of the house on where you think the company is going to go in the face of what are very, very real speculative risks at this point. We're going to cover those as we get into this video as we look to deliberate around a stock price that has demonstrated its ability to move off of its base, moved off the base quick.  When I'm criticizing your ability in your application, I'm criticizing myself as well. We're all retail investors. The difference between us and institutions is worth taking our own personal wealth at a scale significantly less. Yes, but at an impact that can be felt and it can drive emotion. It can really put whatever plan that you put into place to test and it weighs on people. It really does because if I was an institutional fund manager, it would be a piece of cake. It would be relative for me. Like investing $1,000 in a stock is different for me or not so different in the fact that if I was investing $10 million in a stock of other people's money, I would have no emotional connection whatsoever except for my professional obligations to do due diligence on the performance that I'm seeking over that fund. Retail investors need to understand what it is going to take and right now is going to be that time where you have to acknowledge the importance of investing in the face of quantifiable risk. I think the quantifiable risk right now is just about as heavy as it can possibly be. What do I mean by that? Let me put it in layman's terms. If the stock ever in its ability to make orders, make top-end revenue, approach profitability has an opportunity to get to let's say a nice round number, $25 a share, okay? Complete nutter speculation. Let's just put aside the fact that highly on holdings, let's just play for a second and suggest that it will never get to $25, never, okay? But let's play for a second and suggest that it actually ends up at $25 per share. Is it going to be easier or more difficult to own the shares at $25? I want you to think about that for a moment. For many of you guys who have cost averages down below $10, myself, it's going to represent sitting on quite a mass of fortune because you will have weathered the very critical period in the company's evolution of what I would deem to be the most risky. In other words, if highly on starts to turn out a certain predictable order number, they actually integrate with the OEM and the capacity that we think that they can. There's actually a breed over into the freight liner line and actually are able to, I don't know, turn out a predictable, multiple few hundred orders, dare I suggest maybe into the thousands. I think the first year that highly on turns out a thousand orders, I think is going to be huge catalysts for the company. I'm here to tell you, I don't believe that if they're doing that on its first year and it's expected that that order book is going to grow exponentially, stock's not going to be at $25, my friends, it's not going to happen. Stock's going to be well north of that because the stock market will have no choice but to acknowledge that the potential is coming to fruition. The potential now is the very risk factor that entering into the stock now at $3.10 or whatever it closed out on Friday is inherently baked into the idea that highly on is going to fall on bad times. It's going to need to raise funding. Now, I think highly on into strength, race is funding no matter what. I think the stock gets to $6.8 and I think they go through an enormous funding run because I think this few hundred million dollars of capital that they have, everybody is fixated on it. At some point, they're going to need insurmountable amounts more than that to actually get where they need to go and what they need to do, both here domestically and then dare I suggest abroad. We're not even close to those discussions here as we're looking to start the penetration of the domestic market and understand what kind of integration we can get to the fleets here domestically, which I think that they will. Will it happen in 2024? I'm here to tell you no. I don't think so. I do that so people can have realistic expectations and then be surprised to the upside. If something shakes out, something happens, some big order comes through where they've got to jam orders through quicker than what is expected, but that is presumption on my part and I don't think that's going to happen. I really don't. I think it's going to be a slow burn to China. I think Kylian is going to integrate the fleets. I think the fleets are going to be encouraged by the first iteration of the hyper truck ERX, but I think it is going to, by nature of if I was going to just put a decision point on this whole integration piece, I would suggest that it's probably going to be a lot slower of an integration than maybe we initially thought or what we are presuming now to be. I think we should probably prepare ourselves to understand that the integration is probably going to be slower than quicker. What does that mean for the stock price? See, I think with the disconnect and the company's progress and the current stock price, I think we could actually get a little bit of surprise in understanding that is slow as Hylian wants to move or chooses to move or inevitably wants to move because that's what the industry demands that we are going to have a stock price that acts differently. I don't think that Hylian is going to be punished for that. I don't. I think in the last three years in earmarking this company's ability to grow its business smartly, avoid controversy in a red ocean of absolute minutiae sewer garbage that the SPAC environment created that there to be commended. And I think Thomas Healy is to be commended in navigating through these waters in a time that has been non conducive to companies like this to actually navigate and end up where they are now relatively clean. There was no decont decontamination needed for Hylian coming out of this minutiae. They just are where they are moved a little slower, perhaps by their admission for good reason, supply chain issues, global pandemic, etc. etc. The bears would say none of that is justified warranted. But I could play ball just as easily and come up with negatives on a company that I don't want to see succeed, which is a premise that I don't understand. See the Class 8 trucking space has a need. There is an addressable market here in that the wave of momentum that we're actually investing in, in small part in this case, is related to Hylian. But in the larger case, the movement to lowering ESG scores for companies, the carbon scores, the environmental stewardship guide, or the guidance, or the carbon scores that each company turn out is very, very real. I talked about this with the CEO interview I had just last week. I think a lot of new investors are actually investing with the governance to lower carbon emissions profile as being one of the higher conviction pieces as to why a younger generation would invest in a Hylian as opposed to put all their money into Altria Group. And I don't mean to pick on Altria Group. It's a fantastic business, but let's be real, it's killing people. And a lot of people would say, well, money's money, I invest for money, and that's at the bottom line is the only thing that's important, perhaps maybe to our generation, which I think is kind of cynical, but the younger generation, not so much. I've got a little piece of this. When I suggest that Hylian is actually a very, very easy company to invest in, because if we get to enjoy some of the evolution of this young company in doing what it is that they're going to do, the very fact of the matter is that the large looming issue out there in that companies and fleets and customers need to find solutions to the problem of pollution, that the world will be a better place with Hylian rather than out it. So we can parse words all day. We can talk about the stock at $3.10, all we want. We can talk about an apparent strength, yada, yada over the last couple weeks, which I contend just like the stock going down. The stock going up means nothing either. It's nice to see a little bit of reprieve. It is. I've jumped to 20 grand, a couple stacks of high society in my position. I didn't even notice. I'd have to look to see if it's actually one or 15 or 20,000. I don't even know. I don't really care because the material move now is immaterial as far as I'm concerned. Yeah, it's up over 30%. That's great. But we need to understand from a fundamental perspective. It's a deep fundamental. I saw the technicals on Hylian come out and it was very, very interesting to suggest that it's due for a nice little breakout here if it can continue to provide some strength here. But from a fundamental perspective, this story just makes sense. Right now, you're probably stepping in, like I suggested, to a time period where there are the most unknowns. And as those unknowns start to become defined, as we work forward from quarters here, going into the latter part of this year and beyond, that's going to be the key here. And that's all I ever want to see is progress in one word. I want to see progress because from a fundamental perspective, I can move some of these prove it stories or the speculative nature out of the speculation category and into the known category. All right. So let me go ahead and disclose to you guys the four certain stuff that we know CEO stability. I don't think Thomas Healy is going anywhere. I actually think his performance thus far has been very, very good. I think he represents the company well. I think he's well spoken. I think he's smart. I think he understands his business in and out. I think the company would suffer if he for some reason exited stage left or was fired. None of that's going to happen. Okay. I think for certain the board of directors and the upper management is pretty set. I think the churn at the CFO level is just par for the course. I don't really want to read into that. I could the bears are, you know, like why did she leave? I don't know. But all I care about is stability in the current presence and their ability to carry forward on their obligations and especially the regulatory obligation to the SEC and that they're going they're going to do that. So that from a perspective of being for certain is a stability factor for me. The specs, the specs, we have mold over the specs of the hyper truck ERX for a long, long time. Just as of late, the news that it's been provided through social media, discussing the positive results with the hybrid EX makes me wonder if the hybrid EX is really dead on arrival or if it's actually doing really, really good service for existing fleets. You know, I don't know. I don't know. I'm satisfied. I've always liked the hybrid. Some in the community don't, you know, their dollars are invested in the hyper truck ERX. I think from a bridge product to the hybrid and introduction to highly on taking an existing truck and putting some fuel savings to the bottom line, I just think is a no brainer. You bolt it on, you have X number of years to pay back and it just works. It just makes sense in my mind. Some would disagree with me and that's fine. But the specs are fairly well known at this point. The performance of the hyper truck ERX at the thousand mile range is pretty good. The torque we know is fantastic. The acceleration, the quiet ride. We'll talk about that a little bit when we talk about driver experience, but understand what those knowns are. In other words, are we going to be surprised that the hyper truck ERX gets introduced to the rigor of class 8 space and find that it's well, I don't know, broken down on the side of the road, like seemingly every tweet that I get every single day on Tesla. It's amazing. They own Twitter. It's amazing how now I comment on one Tesla feed and I get all these like Tesla is the greatest thing ever and it'll give you a back massage on your way to work and this and that. And it's like, I don't want any of that. So I'm having to block it from Twitter. It's like, holy moly, it's amazing how social media thinks that they can control the dialogue. And for the most part they do, they do. But I don't understand what the attraction is, but I think we're going to enter into the class 8 space having the ability to verify some of the specs that we know now to be true and to be validated when the introduction to the fleet happens. Acceleration, torque, range, certification. Certification is something that we do not have now, but it is in my mind an utter certainty in the first part of this year. If we enter into the Carbonitsa certification, we now have a commercially viable product that currently it does not have. So we wonder why some of these low volume production numbers are coming through the way that they are. I just can't not mention the DSV order. I get it. It's fantastic. I think it was a real solidification on a question mark that's been around for the last three years. Now we can remove that from the speculation category and move it into the for sure thing. I think that's great news. Don't get me wrong. I'm looking to focus on the fact that these orders that are coming in under the Carbonitsa certification are that much more telling if you're willing to pay attention on how willing these fleets are to take this leap of faith, same leap of faith that investors are taking on the onset with this company at the ground floor in the most vulnerable of times of the company, in my opinion. DSV doesn't have to make these risky moves. They don't have to do that. Ruan, Schneider, they don't have to make these risky moves. They don't have to. Warrner, these guys have been working with Ilion for a long, long time. It seems like if I was going to put a scale of one to 10 rating on the satisfaction rate of the fleets that get their hands on these products. I mean, CR England was one from the very beginning. They fell off the radar for me. I just recently saw that it was spotted running their hybrid unit. How good or how satisfied they are with that. I don't know, but for me, it made me think that it wasn't a one off. It wasn't a customer that came and went, but it rather is still using the product. So if I'm going to award a score to Ilion, I'd give them an eight or a nine out of 10 in all fairness. I just haven't heard that much negative. Have you guys? I just haven't. And I'm talking from start to finish, what we know about performance, what we know about the driver experience, I would give Ilion a very, very high mark in the small scale reaction that they've got in introducing the rigs to the rigor. And I think there's a lot more going on right now that we are not privy to. And I would expect based on my deductive reasoning that those feedbacks are, for the most part, very, very positive. And I'm not talking about tweaks. I'm not talking about recommendations or even major things that have been identified that can be integrated, fixed, and put into the final iteration of the HyperTrack ERX once it does become commercially viable. I'm not talking about that. I'm talking about major missteps, major breakdowns, major problems with the generating unit, major problems with the batteries overheating, major problems that were overlooked with the e-axles. Guys, I just haven't heard it. I just haven't heard it. So when we talk about the specs and the certification and the knowns there, I think it's worth acknowledging that I think Ilion has put together a pretty impressive product thus far. Okay. And it's going to be that much more apparent with regard to them focusing on the electric power train itself, as opposed to the entire truck, once they start to really define these relationships that they have with the OEMs and their partners, because all they're going to be doing is producing the power train and allowing the OEMs to do what it is that they do, and that is build an awesome truck. I think that's going to become more apparent with regard to the business structure and the business model that Ilion has ginned up that is unapparent right now, because it's going to be more apparent, the more volume that's turned out as those margins are able to be actually realized on the mass producing of the electrified power train solution, which is what Ilion does. The knowns for certain is the driver experience. We've had some of that in our close community here in the community. We've had a few people, Jason specifically within the community, went down to Austin, actually got behind the wheel of the truck. And there's been a lot of those tutorials of people leaving the cab of the truck and talking about how awesome it is. I'm not making this up. If they got out of the truck and said that was the worst experience ever, I would report upon it. I would. And if I got a flow of consensus from the driver feedback that somehow the experience wasn't a complete improvement from the diesel experience, I would second guess my investment. I don't have that. I don't have that. For me, this is a known category. It's a for certain. Dare I say some may call it a slam dunk when you're bottling up these for certain in your thesis of a company. You have to take it as what it is and be neutral. Don't be subjective with your application, but for the most part, the feedback that I've got from the drivers has been overwhelmingly positive. Again, you can correct me if you've heard differently. That's just my perspective on what I glean of the information that's being provided publicly to me. Initially, what we know is that the cost per unit will be higher. He Leon, who I've been fairly critical of in the past, and I do want to apologize for that. I'm always interested in different schools of thought, different opinions, but he posted a pretty good short of Thomas Healy at Davos speaking about his order with DSV. I thought it was cool. There's a statement right at the end that was pretty snarky, but it was very smartly delivered in that he said, why wouldn't DSV order and everybody kind of laughed and you heard DSV. I would imagine the reps say we will or we would. It's pretty cool actually, but Thomas Healy said, look, the upfront cost of the EVs across the landscape, there's not going to be an EV out there, whether it be a Tesla, Nikola, highly on, that's not going to carry with it some upfront cost. You're going to have to pay for the technology. It's not just going to be about taking a mechanical powertrain and a diesel engine, putting it into a shell of a cab, and put it in the rigor of over-the-road trucking to be subject to their maintenance cost, and most importantly, their fuel cost, which segues into the next statement on the hyper-truck ERX when we're talking about the initial onset cost is $180 from the status quo. Diesel onset is fairly low, but as you drive the vehicle, the longer you drive it over to its expected life, the cost is what's going to drive up the actual overall total cost of ownership. It's just different. You make those initial investments into the tractor upfront, and the expectation is that by running C&G or R&G into that rig, even with the fuel credits, even some of the upfront cost incentives or incentive to the cost, the $40,000 credit, which I think is going to be negligible. I think it's okay. It's great. Companies know they're going to have to move in that direction, so it's better to have than to have not the incentive, yes, but when all of the dust settles, the upfront cost is going to be higher. We've got it around $400,000 per unit per cost per tractor. The idea is that if a fleet takes on five units, 10 units, whatever it is, what the total cost of ownership is per unit over the time frame that that unit can be introduced to Class 8 Rigger times the number of tractors in the fleet is really what it comes down to. Can they run longer? I don't know. The longer the Hylian Hyper Truck ERX runs, the better it is for fleets, the better it is for the environment, the better it is overall. Okay. Thomas Healy just puts it out there and says, look, the upfront costs are going to be more. It's to be understood. New technology comes out. There's a lot of technological advancements in the Hyper Truck ERX. It's not just your standard diesel Class 8 truck. It's got some improvements to it. Some of them are intangible benefits that cannot be quantified on the bottom line, like the next one that we have, and that is driver experience. Driver experience, like I suggested, is that quantifiable? Well, I don't know. Is it more attractive to drive the Hyper Truck ERX for a new generation of drivers coming into the business now? I don't know. That's yet to be seen down the line, but I think the cost upfront per unit being higher is one of those givens that we just need to talk about. We need to acknowledge and we need to identify as we march forward in the evolution of the company to see how they can drive down that initial cost and how quickly they can get to that payback level and get to that level where it's actually paying the fleets back to run the units. Okay. Now, with that said, the guarantee or the known at this point to the first certain is the low cost of fuel. Okay. The cost of fuel being lower on the C and G R and G side compared to what it is for diesel at around four or five bucks a gallon is it's significant. And the fleets know this. And they know that in, I presume the fleets look at it this way. Look, diesel is not going to be going away. Diesel is going to be around. Diesel is probably going to in some of the routes be the perfume preferred fuel a choice for that said route. I think what highly on allows them to do is provide for some fuel agnostic application in that they're not just subject to even if they are running C and G and the fleets, the highly on hyper truck ERX may give them the opportunity to in some small capacities explore new technology and they may have those routes picked out the longer haul routes where the ERX fits well and therefore highly on offering the product that they do will benefit from that and the fleets hopefully can find that cost of ownership benefit and not be subjected to what was a high cost of fuel over that same route when they introduce the hyper truck ERX to that route. All right. But for certain benefits when I talk about the portfolio of products that highly on has, I don't think we talk enough about the the cloud application, the algorithmic and the on board maintenance and and the predictive maintenance that they're going to have with their technology. I'll let highly on take that and evolve that in time. I think they've got a real winner there. It's been discussed over the last three years. It's not something that gets discussed very often through social media circles. Unfortunately, I think that's a miss. I think the ability of a company from a holistic logistics and maintenance protocol posture can provide that algorithmic data in such a capacity to predict of predictably identify the maximum efficiency of their fleets by using the software. I think it's huge. It just makes sense and for people who would suggest that we're somehow advanced in our application and using cloud application and algorithmic data and the extrapolation of that data to turn back real work, real value. In other words, do we have by nature of statistics and understanding of what is the realistic shelf life of the battery in the rigor of Class 8 space? Two years, three years, five years, I don't know. So based on what the data from the truck is telling us that we are all ops normal on the battery or if by nature of us approaching that end of shelf life that we can cap that and we can predictively put some maintenance on there for a module change out because the degree or the statistical inferences that we're approaching a time where we could have or the potential for a catastrophic failure increases, therefore we cap it and we go with a safer application in and that application goes with anything on the truck. No matter what the shelf like of is of the components, what the periodic safety checks need to call for on the trucks, depending on what service it's introduced to, there's going to be different areas of focus in cold weather application than there is in warm weather applications. That data is going to be invaluable and as Hylian starts to capture that data and really starts to quantify and really sink their teeth into what I think they're going after in this is not just the powertrain, but look if you introduce the powertrain in a way that we want you to, we can predictably get the maximum amount of efficiency out of the vehicle whereas before we were significantly much more reactive. Now, I'm not to suggest that there isn't manufactured recommended specifications on diesel maintenance, etc. However, this is just a different way of thinking about those recommendations coming in proactively rather than reactively. Okay, and we get over to the prove it side of the house and this is where the real risk exists with the company. This is where I think new share owners are going to be tested. This is where I think if you're going to be an investor in this company, I think it's going to be one of those things that is going to be long standing and we're going to have to allow time to really address some of these things. Okay, so this is the four speculation category. Your list may be different than mine. These are some of the ones that I jotted down that I think if I was just going to push it out into the future and say here are the things that highly on really needs to answer some questions around. This is where it starts for me. Commercial rollout. I have questions. Some in the group, I know Excalibur is pushing out some really good content and shout out to Excalibur. I really like the work that's being turned in right now by Silent Alert. It's enjoyable for me. It's great if we could have more than a handful of people pushing out highly on content. I think that would be fun. It's worth it. It's a lot of fun, man. Wall Street engineer as well. Really, really good sober trades as well. I know he's always part of the community, always tweeting and always in the Discord group fairly frequently, but we all have some different ideas about how this commercial rollout. I think Excalibur kind of brings me down to earth sometimes when I listen to his video because he's very deliberate about his process. Honestly, if I could give a compliment, there's a little bit of that pedigree in stock market investing that people need to resonate with. In other words, you can't get all wiry when you buy a stock and the stock goes down. With that said, you can be critical. You'd rather see the stock go up and it didn't go down. I'm not saying be happy about it, but your application should be somewhat deliberate. His twist on the commercial rollout is a little bit different than mine. I'm a little bit more skeptical on how they're going to make this happen. I've heard during quarter's call before that the idea here using OEM mod centers to get their product started, get it in the hands of their customers, and then get on the OEM line has morphed into Hylian's going to be taking delivery of the chassis, building the trucks in Austin, and turning out the trucks to customers. Boy, oh boy, if that becomes a long-standing thing, this is going to be really, really disappointing because that is against the business plan that was initially presented. The Austin plant does not have the space, nor does it have the assembly line and technology and the quality control checks necessary to build trucks in Austin. When I heard this, I was very disappointed. Can they do it in small volume? Hell, I guess, but Austin is not an OEM. That manufacturing facility is at best a warehouse for assembling and doing R&D on their current products. I don't know what goes on on the inside there. I've only been flown through it with a drone, but it's not an OEM. It's not an assembly line. It looks like a garage from the best I can tell, and this is no knock on the Austin facility. It's just a sheer reality that I don't think they're going to be able to turn out volumes of any material benefit at all, and I don't know how they're going to assure quality assurance by turning out of what I'm sure is subjected to rigor at the Peterbilt plant. The Hylian facility is not subjected to that rigor, health and safety inspections, health and safety of workers, the quality control checks that are in place. If I'm a customer, I'm not taking my truck's delivery from the Austin facility. I'm just not, and there's going to be bulls of the company that don't like me to say that. That's the reality. If I'm DSV, I want my stuff coming off the Peterbilt line, and I'm going to provide the leverage back to Peterbilt to say, whenever you're ready to build me my 10, I want it to come from the Peterbilt facility. I don't need it to come from a warehouse in Austin and be delivered to me with a bow on top. I don't need that. I need the assurances that it's coming off of an established OEM line. Okay. So the commercial rollout, we will see what happens with that. Another four speculation issue is along the lines of what Tesla is incurring right now with the inability to deliver on durability. This looks like a monkey circus. The Tesla rollout? Yeah. Yeah. It's really just like a seal bouncing a ball on its nose. I can't give any proper credit or credence to what it is that they did in their rollout. They can get away with it. They're Tesla. They can do anything they want. They can tell anybody anything that they want. They can sell their product to anybody without any promise of any type of return whatsoever. They can recall thousands and thousands, 10,000s of vehicles. They can trap people inside and it doesn't matter. Tesla can do no wrong. Okay. There's some people actually suggesting that Tesla is a buy right here from a technical perspective. Good luck with that. I'll own Tesla through the indexes because I have no choice in doing so. But as far as owning single stock in Tesla, I will not be playing ball. Durability. In the case of Hylian, they have to be able to execute along an expected timeline of the tractor with some level of predictable durability. Is it going to be subjected to breakdowns all over this country when it does end up in the commercial fleets? And we know that they're highly on units and they're breaking down with some level of frequency. That can't happen. That can't happen. And one thing that's missed all the time with what diesel has delivered over the last 100 years plus is durability. I'm a believer in diesel engines. I am around them on a daily basis. And if you doubt the ability of a diesel power plant to deliver power, you would be sorely mistaken. Both on the generating side as well as the propulsion side. They're phenomenal. How is Hylian going to stack up against the durability? This is a real unknown for me, guys. And a lot of people are saying, well, Ryan, they just did summer testing. They ran up a hill.  Fantastic. Mean squat to me. The real test of the rigor is after certification, after Hylian has done everything that they can possibly do to burn up years in validation only to introduce it into the rigor of class eight to fall on its face and subject itself to major breakdowns. That is going to be a black eye that I don't think they can afford. I don't think they can absorb. And I think it's something that they need to avoid like the plague. And once we start to get some indication that durability is there, we can start to slowly shift this over into the first certain category when we understand that we've gone through an entire cycle of the Hylian hyper truck ERX first iteration. What were the learnings from that? What were the cost savings? What specifically were the durability? What were the hiccups? What were the identifiable breakdowns? Because there will be, there will be. Okay. And we'll move on from there. The prove it in the cost payback to fleets. I know when the numbers are run that they're very attractive and yada, yada. I need to see that proven out. I need to see that fleets are actually sitting on earnings calls and actually by name discussing the Hylian bottom line savings. Is that a pipe dream at this point? Yeah. It's inherently where the speculation exists because we may very well may just be talking about an equivalent to diesel and that people believe in the Hylian story enough to where they can ship goods from point A to point B, but there is really no cost savings. The upfront cost absorbs what is typically spent on diesel on the back end, but let's say diesel gets cut in half. Let's say diesel goes back down to $2 or $3 a gallon. Is that going to happen? I don't know. Could it? Yeah, it could. If OPEC decides that that's what they want to do, yeah, this entire country is run by the mafia. Okay. If that's what they decide to do, then Hylian's cost savings isn't going to be so savory right now when compared to $5 diesel as opposed to $4, $3 and so on. Okay. What if we have a regime change from a political perspective and now the renewed focus on drill baby drill instead of conserve baby conserve? I don't know, guys. Okay. But the total cost of ownership and the payback for fleets is something that I want to see quantified on the bottom line and I just don't think we have it. I know there's people who have run the calcs on it.  I want to see that on paper and until I see it, that is going to be a proven story for me with the potential of shifting into the for certain category as we march forward on this opportunity. The adoption phase, the adoption pace, excuse me. It's something that I talked about. If you think that somehow you're going to be delivered this silver platter of phenomenal news at the end of 2023, I'm going to go with my gut on this one and an inability to provide performance metrics and suggest that if your expectations are overblown, you are probably going to set yourself up for disappointment. I think the proven story for me in the adoption pace is I'm setting it up for it to be slower than fast. If it's going to be at a faster pace and I feel like, okay, we've got an accelerated pace of orders coming in, we pick up a few hundred orders here, you know, 50 orders here, a few new fleets step in. I don't know. We expand the innovation council. Why not? Then I will eat crow. I will say, all right, I'm pleasantly surprised at the accelerated pace of adoption of the hyper truck ERX. And perhaps maybe I'm underestimating everything that I know about demand, everything I know about the time being now, everything I know about the mandates coming down the pike, everything I know about the mandates affecting both fleets and OEMs, okay, to suggest that even in the face of those mandates and incentives, that the rollout will be slower than we anticipate that it will be. Okay. So it remains a proven category. I will update you on those specific attributes of where I think the company specifically is right now. And I think coming into 2023, we probably have ourselves set up against more potential for disappointment than we do for, oh my goodness, I'm so excited. I'm the best investor ever. I invested in a ground four company, and now it's at $50 a share, okay? I don't think either one of those two things are going to happen. I think we're going to have metrics to look at. I think we're going to have metrics to extrapolate around. If we get some appreciation on the stock price, I think it's going to be a give back because I think we are at, if not we have been treaching through the most difficult time if the company is going to be subjected to difficult times, that the company will ever be subjected to. Look, if they start to materialize on some of these promises, we're going to look at these times and laugh, okay? If in fact that potential, if it doesn't go away and it doesn't remain static, if it actually goes up in value because the company has actually delivered on its promise to integrate and be adopted by the fleets with everything that we know, okay? I think that there's a potential for it not to materialize the way that we're envisioning now in our minds as somehow this massive ramp up to commercialization is somehow going to be a catalyst that drives the stock to the moon. I could be wrong. I could be wrong. I will come on and calmly suggest that I'm wrong, okay? No problem. While my bank account increases in value, you know, five, 10, 20 times over, it's okay. That part of it's a byproduct of my application. I'm more interesting in looking at the fundamentals of where we are now and actually massaging over this idea of being an investor in the company. That's what we need to do right now, okay? But I'm merely suggesting that the setup here doesn't look good based on the track record that Hylian has put forward thus far. I don't put a lot of faith in this company to under promise and over deliver. I put a lot on them to under promise and under deliver. Maybe over promise and under deliver thus far. Is that a knock on Hylian? You can take it for what you will, all right? They've been able to demonstrate their ability to do what it is that they've done up to this point. That's got to be good enough if it's not, don't invest in the company. Very, very simple, okay? The adoption pace, the corner technology, it's a big unknown for me. It does go into the proven category. There's going to have to be some R&D that goes into this. Panzer talked about about $100 million investment into the corner technology to get it to where they need it to be. This is a real unknown for me. Do they have $100 million just to put into R&D for Carnot right now? No. Do I believe they'll do it? Yeah, I do. I believe that they believe by whatever metric that they're looking at that the Carnot is going to be a game changer. I believe that it could be. But I just think with a few hundred million dollars in the bank, I think this company is flirting with disaster. If they're not more fiscally careful, I would not deem Hylian in their application of spending money to be fiscally responsible. I wouldn't. They're burning through capital at a rate that is far outpacing their ability to generate top-end revenues.  I'm just being real, okay? And it's getting to the point where it's almost too close for comfort, even at $400 million. Some bulls would disagree with me. Ah, they're fine, right? No, they're not fine. They're not fine. Not if they are trying to traverse this bridge between very little to no revenue, to actual material revenue to where they can cover OPEX and CAPEX. I don't know what to say. So how they fit Carnot into the equation for me is an unknown. It's a proven story. I'm excited with the technology. Absolutely. But how they integrate that into the business plan over the next five years is going to be of interest to me. And I'm an investor. I'm excited about the Carnot technology. I'm excited about being on board with the GE folks up there and the intelligence that we brought in in-house. I think it's all fantastic. I just think that this company is vulnerable. And it's at the beginning stages of its inception, and they need to start to get some material stuff in the churn. And I believe that they will. But until then, it's going to remain a proven story. The last thing I'll mention is the Hybrid EX. The Hybrid EX has been kind of an unknown. Thomas Healey admitted to some competition being out there a couple of years ago and that the Hybrid EX would struggle to find its footing. I don't see that. However, it does remain in the proven category. There's a lot of bulls. Rick is one of those that really is, he's really sold on the ERX as well as I am with regard to its potential impact on the Class 8 fleet. I don't want to underestimate the Hybrid EX, but I also don't want to dismiss it. And for right now, I think it falls in the proven category as being one of those products in the product portfolio that Hylian has on whether or not it remains material, whether or not it remains a supplement to the top end, whether or not it is improved upon, whether or not the findings of the fleet are coming back so positive that they can't disestablish the product. Only time will tell. I think it'll be fun. But for now, it remains in the proven category and the four speculation. Guys, I appreciate you tuning into this weekly update on Hylian.  I really do. Hopefully you picked up on some of the between the line stuff. This story will unfold, but it's probably not going to unfold in the manner that you expect it to unfold. Get over yourself. Like I talked about in other applications and immediate satisfaction and fixing problems the way we do in all aspects of life, you will get a pulse of investment education from me. And that is to be an investor is a continual definition of what that means. Have fair expectations around where you think this will be. Buffer the trade in a way that allows this investment to materialize with as little emotional content as you can possibly deploy because it's not going to do any good. This is a speculative aspect. We have some fundamental aspects and I wanted to separate those in this video as we move into 2023 because my bottom goal is to really condition investors to do something atypical of the retail investing community. And that is to deploy strategies that are in line with fundamental decision making and not emotional decision making. When it goes to six, you sell. When it goes to 12, you sell. When it goes to 24, you sell. When it goes up to 42, you buy more because you panicked sold all the way up to 42. Make a decision that is atypical of those traditional retail investors that buy at the wrong time and sell at the wrong time as well. Guys, I appreciate you. Leave your comments at the bottom. Subscribe to the channel, hit the notification bell, share the message with those highly on folks out there that are also sharing and monitoring the story going forward. We will continue to provide this product as we do through the YouTube channel and we will be there for the next one. Guys, thank you so much for tuning into the message and good luck in your investment future.


Trajectory for the rest of the course here. I guess what is this April 30th now? Okay So We've got two weeks and then finals that next week and the way that you see siesta's finals I'm so pretty new here I think that finals week there aren't any classes that that week at all So really we have these two weeks to finish everything up So all I'm trying to do is do chapter five the two parts of the chapter that I think are relevant And useful and that just means two sections and one of them I'm going to get done today for sure So all we have to do basically is finish up one more section in review and that that's it. Okay So Having said that I'll start giving you some idea of what to study for the final You know what the final is going to be comprised of in terms of old exams and definitions and all that kind of stuff So you have some idea where to begin I'm probably not going to do that today, but try to remind me if I forget We'll start talking about that on Thursday. Okay, I think you have plenty of time to get an idea of what what to look back over Okay, sounds like Okay, all right So Yeah, so basically here's what we're gonna do it's nice and compact this time Let's see Okay, sorry Joe. I didn't mean to close the door on you there This is Oops See that's what I need.  Okay There we go Section 5.2 5.1 is just a history lesson about there's a geeky guy And the ma who got all excited about all this stuff and just like all these guys back then, you know They died of tuberculosis when the time of their 26 They don't tell you that though So there's my augmentation to 5.1. And so we're gonna move on just to 5.2 for ma's Little theorem is what this is called. He didn't have any pets. Nobody loved him. So he just Throwing it calling it little made him feel better Okay so Actually, yes, I mean there's he's he's actually in the number theory community sort of famous about there's another Theorem called for ma's last theorem which maybe some of you have heard of before that was a much much It's it's much bigger than this theorem. So hence. This is the little one And it basically just says this just says let P be a prime a and integer and suppose that P does not divide a Okay, that's that's the assumption. So this is very important that you don't forget this. He does not divide a Then I'm gonna okay. This P's gonna cut this off a to the power p minus one is congruent to one mod p The statement's very simple. In fact the proof is actually pretty simple And so what this says is of course you can already translate this yourselves at this point and just says that if you take an integer a And piece this is important that p does not actually divide a p's not a factor of a Then if you raise that number to the p minus one power and subtract one p divides that number Okay So this is the first kind of I don't know I think sort of surprising Results in the book, which is why I really wanted to get to this and after we do this Just do a couple of examples for you and then I'm gonna do you know like I've been trying to do some some stuff related to your homework So what we're gonna do is just suppose that again, I'm not saying that What where these things live, but we're just gonna suppose that p does not divide a of course a is an integer He does not divide a and Let's consider the integers a 2a 3a and just we're just gonna keep on going down until we get to P minus one times a okay, so here's here's the idea This is actually not very hard the proof of this theorem is not is not hard And this is the main object that you're gonna be looking at in here Okay, so I'm gonna make a couple claims so claim one Yes, yes Well What okay, so if I say claim what what I really mean is I'm sort of I'm proving something for you a Supposition depending on the context could mean this is one of your hypotheses in the statement of your theorem All I'm saying is when I say claim. I'm just going to show you something that asked it That's true, and then we're gonna use it later on Okay, so the first claim is that none of The above Integers Okay, just to be clear. I'm just gonna make a little box here That's what I mean by the above integers. I mean all of these guys right here none of the above integers are Congruent to zero mod p. Yeah, sorry Joe a 2a 3a all 4a all the way down to p minus one times a Yes, yes, none of these are congruent to zero mod P Yeah, it is it's saying the same thing I'm just gonna use this congruence language because it's gonna come up here in a second. But yes, it's exactly the same thing so Why is that true? Well Suppose one of them was congruent to zero mod p. Let's say suppose You know I have to do this in general now, okay, but let's just suppose that I a is congruent to zero mod p where high as an integer right and Well, what are the possible coefficients in front of the a's right? There's the first one is a so I could be one and the biggest it could be would be P minus one right? You see where I'm getting that Just comes from the coefficients in front they range from one to P minus one well then as Nick pointed out I a congruent to zero my P just means that P divides I times a right. That's what it means. Okay? P divides I a All right, so we're trying to get a contradiction here and so Yes I Say that again You basically yeah So the way to say it is our supposition is that P doesn't divide a and if remember if a prime divides a product of two integers It has to divide one of them, right? So since it doesn't divide this one it has to divide this one But there's no way it can do that because it's too small in some sense, right? It's between one and P minus one so it couldn't be divisible by P. That's the idea Okay, since P does not divide a I'll write this out We are forced to conclude that P divides I because of the property of primes that we've known for a while But this is impossible since Like I said, I is sort of too small one is less than or equal to I Which is less than or equal to P minus one so that can happen, right make sense Okay, so we got a contradiction that means what we assumed What's true is actually not true. So this proves the claim Oh, okay So I is less than or equal to P minus one and oh you're saying is the other part we're done And that one is less than or equal to I you mean Right, right. Yeah, I mean that yeah, you're right, but it just sort of Yeah, I mean and that's that's a good question is what would do you leave out I mean how obvious it does have to be before you actually don't state it, you know Yeah, I mean in this case It's it's just again because it's positive and it's smaller than P So P can't that goes back to a theorem way back in chapter one I think but but yeah I mean really it's the it's the fact that that I is positive and strictly less than P So P cannot divide it because they're they're both positive okay, so claim two is that none of the integers in Box hopefully this makes sense. So I'm writing on a slant here Are try to correct that here congruent mod P. Okay So let's again, and I'm running out of rooms. I'm gonna try to get this in here. Let's suppose not let's suppose that say, you know I times a is congruent to J times a mod P Okay, well, okay Then and now we're gonna actually use something that we've already known To shorten the proof a little bit the book I don't think went into this but I'm gonna I'm gonna quote this because this is something that you should have in the back Your mind to work on problems. This can certainly help you out What can we do here? Well, there is something we can do to this congruence to simplify a little bit What's that? The a's yes, and the reason for that is because A and P are relatively prime right if P does not divide a they have to be relatively prime They have to because otherwise think you think about it Okay, if I'm missing anyone if P doesn't divide a is an integer P and a are have to be relatively prime So what's what's what are the only possibilities for the GCD of P and a one and P because it has to divide P? But it can't be P because then P would divide a and we're assuming P doesn't divide a so it has to be one right So that's why they're relatively prime and so by and I'm not gonna keep writing this down But hopefully you're all with me now Then we can cancel as Nick said we can cancel the a out Okay, and reduce this congruence. I'm not gonna quote the theorem now, but Okay, so then I is congruent to J Mod P Okay, and this I'm gonna gloss over just just a bit because I'm not that interested in it So what does that mean? That means that P divides I minus J Okay. Well, what do we know about I and J? So I And J are between one and P minus one and just just try to think about this I'm hopefully if you just maybe think about this geometrically this will make sense I'm not gonna write out too much here because I feel like I'm starting to split hairs now because I want I want to really get to The more complicated stuff here. You've got two different numbers I and J that are both between one and P minus one What happens if you subtract them? Well? Okay, let's just suppose that I's bigger. Let's just suppose it doesn't really matter But I is bigger than J and they're both between one and P minus one when you subtract them you're gonna get something that's between Zero strictly between zero and P minus one when you subtract those two numbers Okay, if you think you think about it. I mean just just think you know two and seven well What happens when you subtract you're gonna get five? okay So you're gonna end up with something that now regardless one, but maybe the the smaller one comes first You know, maybe you're taking two minus nine or something like that But the point is hopefully you'll buy this an absolute value because I and J are squeezed between one and P strictly Well between zero and P and strictly between zero and P When you subtract You're an absolute value.   Let's just suppose one is less than or equal to I Which is less than J? Which is less than P right? Well, what happens if I take I minus J? You know We're not gonna worry so much about the The one minus J here, but well, okay Let's just subtract and we can certainly subtract J from all these right so One minus J is less than or equal to I minus J, which is what? less than zero Okay, right, which is less than P minus J and actually You can show that I minus J then has to be between zero and minus P So again, it's just it's too small to be divisible by P. So that's all I'm gonna say about that Well, okay, if you go back over here So J Okay, I'm just answering this question J is certainly less than P plus one right for sure That's for sure, right? Because J is some number between one and P minus one so Let's see That's not what I wanted to say. Sorry so So what do we know about J? So Yeah, yeah, okay. So no this this is all this is all amounts to so J is less than P plus one. Okay, so From here we get Yeah, so Minus if you move everything over here you get minus P then is less than One minus J. Okay, I know this is getting squished This is definitely true We're mad J is definitely less than P plus one because J is one of these numbers here Okay, it's actually less than P even but Okay, but if J is less than P plus one then just move the minus P over here and the minus J over here And you get minus P is less than one minus J Then once you've got that then you've got minus P So so then you've got I minus J squeeze between minus P and P Okay, so the only way to be divisible by P is if it's zero that's that's it. Yeah, sir It would be sufficient to say if I and J are Either I equals J in which case no problem or they differ by a factor of P and we don't have that many But that's exactly the thing. I'm trying to prove rigorously here. I'm trying to answer his question Yeah, I think you'd still need to do to demonstrate that that too which is what I'm trying to say here And this the minus P less than then one minus J is Where I throw this in here so Minus P is less than one minus J and then you've got one or sorry You've got I minus J strictly squeezed between minus P and P. So it's got to be zero. Anyways, all right No, I don't want to focus on this anymore, but that's the idea Okay, I go on to a new page now Okay, so So now the thing to note is that This is kind of the cool part of the proof the integers a to a three a on Down to P minus one times a non-zero, right? I showed you that none of them are zero mod P and Mutually incongruent So what does this say? First of all if you take all the integers mod P how many distinct Integers are there reduced modulo P. They're exactly P of them right zero one two three all the way down to P minus one P's congruent to zero right P plus one is congruent to one. They're exactly P of them So I just showed you that none of these are zero and they're all mutually incongruent mod B That and in other words neither of them is and I showed you this neither of them is congruent mod P and so A has to be so listen what I'm saying. I'm just going to say this and I'm not going to write it down Because it's just going to get too long a has to be congruent. It's certainly congruent to something mod P Right, so every integer is congruent to something mod P. It's the remainder when you divide it by P, right? A is congruent to something mod P That's not zero right Two ways also congruent to something mod P, but it has to be congruent to something different mod P Otherwise those two guys would be congruent and I just prove that they're not so the point is all of these guys all of these P minus one Scalar multiples of a are congruent to different things Mod P and none of them are zero Because I showed you that none of them is congruent to the none of them is congruent to each other mod P So I'm just okay. I'm just going to say this too. There's a theorem. This goes back into chapter 4 So let's say that a and 3a were both congruent to 5 say mod P then there's a theorem that says that okay, well a is congruent to 5 and 5 is I can't remember a and 3a okay, so a is congruent to 5 and 3a is congruent to 5 So we've got a is congruent to 5 and you can flip the congruence around so 5 is congruent to 3a and by transitivity a is congruent to 3a, which is what we just proved isn't possible That's why they all have to be congruent to different things Okay, so what do we know and I want you to think about this for a second All right, this this might require just a little bit of thought, but I again I don't want to be too wordy here a times 2a times 3a on down to P minus 1 times a is congruent to 1 times 2 times 3 times 4 all the way down to P minus 1 Hang on a second Mod oh, that's that's bad. Okay I'm gonna try to erase this, but I'm not going to Mod P there we go. All right Sorry about that. Well. Yeah, I mean So the point is I think is what you're making here It's not necessarily the case that a is congruent to 1 and 2 a is congruent to 2 and 3 is congruent to 3 That's not necessarily the case, but here's here's what you you should keep in mind and this also comes from a Induction in an application. I think I think it was there in 4.2. At least it was in section 4.2 So, okay, let me let me just say this Remember what I said before a is congruent to one of those numbers for sure It's not zero remember none of them are congruent to zero mod p so a has to be congruent to one of those guys Maybe it's not one. Maybe it's Seven who knows right? Two ways congruent to a different one of those guys as we were just talking about right three a's congruent to a different one of Those guys how many how many elements are there here? They're p-1 here They're p-1 elements here, so they all match up in some way Okay, they all match up and so because they all match up Remember this is also a theorem if a is congruent to b and c is congruent to d then ac is congruent to bd You can multiply them together. That was also in theorem 4.2 I believe so once you have these these individual congruences you can literally just multiply them down the line And that's essentially what we're doing and we're just using the fact that Multiplications commutative just to write it in the natural order on the right-hand side Thus, so let's let's rewrite the left side. Okay. There's a natural way that we can rewrite the left side All of these terms have an a in them, right? There's p-1 of them, so we get a To the p-1. What do we have with the numbers? Well, okay, you see what I did I just take I just took all the care of all the a's but now we saw two times three times four times five Down to p-1, which which is p-1 factorial, right? You might say well, but there's no one But it doesn't matter if there's a one right if you leave the one out and where you put the one on you get The same thing because you're multiplying by one. Okay, and then of course on the other side we get P-1 factorial Mod p. Okay. We're almost done and What can we say here? Yes We can do that. Yes. Why can we do that? It is relatively prime to p, right? Okay, and it's it's not because It's not as easy as saying that that p-1 factorial is smaller than p that may not be the case I mean if you take five factorial right it it's or five minus one factorial. Okay, four times three times two times one That's not smaller than five. It's still bigger, but you know that Okay, so since somebody I'm just right quizzing you today here, but somebody tell me why it is That and remember in order for this to be relatively prime to p All we need to know is that p doesn't divide it then it's automatically relatively prime from what I was saying just a little Bit ago. Why can't p divide p-1 factorial? That's basically the idea Right, right, right. Yeah The point is if p divided p-1 factorial remember it's a product right p-1 times p-2 all the way down to one So p is being prime when has to divide one of them Okay, but if p divided any one of them that's not possible because that factor is smaller than p So p camp so sorry So p can't possibly divide it because p's too big right whatever that number is p's bigger than it so p can't divide it Okay So we can now cancel the p-1 factorial to get a to the p-1 is congruent to one Mod p okay, and I don't want you to forget the assumption in the beginning of the problem was that p does not divide a Remember, that's very important. Oops. That wasn't supposed to be a D. Yeah, sorry How about that? Mod p. Yes Sure, let me just let me kind of do this as an aside over here Okay, this I think we'll make it make it simpler. Hopefully I gave myself enough space so For example, why can't you know five divide a five minus one which will be four right? Five can't divide four factorial right? Well, this is the same thing as saying five cannot divide one times two times three times four right if so Because of this product remember five will have to divide one of the factors That's one of the properties of primes is if a prime divides a product that has to divide one of the factors So five would have to divide one two three or four which it can't because it's just too big right So then five would divide I for some I between one and four which is not possible Okay All right Hopefully this is legible here. This is really horrible P. But yeah So that's what the theorem says and this is a kind of an interesting trick here If Yeah, if we if we had any actual real college bars here You could you could impress your friends by whipping out your calculator and saying hey, guess what I can I can I can show you that you know 13 To the 102nd is Divisible by minus one is divisible by 103. I'm like, how would you know that and they can ask you these questions and seem like a genius It's not gonna really no one's Now if it was that easy believe me I would I would do it but doesn't it doesn't work that easy So that now but in actual I mean in all honesty it is kind of cool that this is true You take a number that's not divisible by P and for some reason if you raise it to the P minus one power and subtract one It is divisible by P. That's a sort of a non-obvious fact Okay, I see well it sort of Sort of falls apart right away Because If you took that away then for example just plug in P for a here then certainly P It's not the case that P doesn't divide a P does divide it But then P will divide a to the P. I mean just just imagine replacing it with P Okay, then P does divide a in that case Then P is definitely not going to divide this because this will be already be a multiple of P and then we need to subtract one There's no way you'll divide it. Yeah Okay, and that same argument actually can be used for any any value of a that's divisible by P You just replace it with Pk or something and you can see that it's not going to divide that to the P minus one minus one Okay, so the corollary to this and see I may not get as far today as I wanted to but that's that's okay I mean I'll basically get at least get through this section So then we're going to have plenty of time for them for the next section and review So corollary is actually very nice very succinct very easy to state and that is that for any integer a Let's see. What is the corollary? a to the P is Congruent to a Mod P yes For any integer a and you can try some test cases here and just plug in the simple ones like zero and one And you can see that they both work, right? In fact, it works for everything Peace prime still. Yes. He's still prime and so the proof here is really pretty simple. So the proof goes like this case one P does not divide a then By I'm going to abbreviate this. Hopefully you don't mind Flt for ma's little theorem, right? a to the P minus one Is congruent to one mod P, right? Okay, that's what we just proved So and this is also something in this you should have known going to the next exam You can always multiply any congruence by any number you want and preserve it. Okay, so hence Multiplying through by a that's you want to write that down. I get a to the P is congruent to a Mod P Okay Case two P does divide a Okay well Now we're just gonna What is it we need to show I just want to remind you what you need to prove we need a To the P is congruent to a mod P Okay All right now you might think I'm being lazy, but I'm just gonna write obvious here, and there's a reason why do that Here you can write it down you can write it down. I'm gonna tell you okay. I'm just running out of room And I want you to think about Think about this. Okay case two is that P divides a he's a factor of a If P divides a Do you think that P is going to divide a to the P? Okay, just just think about what this is just think about it written out a times a times a times a times a p times Well to replace every a with pk right pk times pk times pk times p certainly going to divide it for sure And if P divides a then P divides a and Therefore P will divide their difference if it divides both things it divides their difference done That's it. That's all there is to it. Does that make sense? Okay, I didn't want to write all that out, but hopefully that's not too hard Okay, so now I'm gonna do an example and You know probably what I'll get to Please tell me if you need a little more more time on this before I go to the next page Okay, so I'm gonna get give you an example that has a flavor of of your homework, and then I'm gonna do one more Lama which might be used in the homework. I'm not sure but just to be on the states out I'm gonna go ahead and go through it or at least I'll tell you what it is I mean I prove it, but it's the proofs in the book okay, so Here's an example to verify to the 38th power is congruent to for mod 11 Okay, now the direction is actually going to say more I'm like used for my last theorem to do this Yeah, sorry not for my last theorem. Definitely not for my little theorem. Yes Okay, so Here's the thing to do and again that the principle is very similar in a sense to what you were doing before if we want to Okay, we want to use for my little theorem to do this. Okay, so what's the modulus? It's 11. So remember what for my little theorem says it says that p doesn't divide a then a to the p minus 1 is congruent to 1 mod p So that knowing what the p is and what the power is should be immediate. It's Whatever the modulus is that's what you should be starting with. I mean at least that that should be your first strategy Okay, so and I want you to think about this. Okay, five to the 11 Minus 1 is congruent to 1 mod 11 okay, so the a is five right the p is 11 and P does not divide a in other words 11 does not divide five Right, I'll write that down if it helps The a is five in this case right the p is 11 and just know it. I'll just put this in brackets P does not divide a this case right, okay So What does that tell us? Well, let's we can write this a little bit nicer, right? So this is 5 to the 10th It's congruent to 1 mod 11 38 okay. Well. Yeah, we're gonna try to get there. That's our whole goal is to get there Okay, and you'll see you'll see how this works here in a second This is this is just one way of going about doing it. So we've got 5 to the 10th We want to get up to 5 to the 38th right so Well, what can we do the nice thing is that and of course as you remember what our goal was is to get ones popping out We've got one on one side so whatever we if we raise first of all we can raise both sides Whatever power we want and the right side is always going to be one for sure and we want to get five to the 38 so It stands to reason maybe that The okay, and this is the way you should be thinking you might think I'm gonna just raise both sides to the third power That'll give me up to the 30th But you still have a lot more powers left you're better off raising both sides to the fourth power Because five to the 40th is a lot closer to five to the 38 and then work from there. Okay, so And I'll write this out so five to the 10th to the fourth is Congruent to one to the fourth mod 11 right, okay All right, so now we've got five to the 40th Congruent to of course, I don't need the fourth power here one mod 11 Okay. Well, what can I do from here? I'm kind of bigger than I want to be Okay, I want I want to know what five to the 38th is so what I can do though Since I'm not that far away what I can do is just take five to the 38 out and then I've got a five squared left, right? Make sense Okay, so we're almost done Okay, so I'm just gonna do this in one step here. Let's look at the five squared is 25, right? What's 25 mod 11? Three right right sure. There are other ways you could do it too So now let's look at where we're at Where do we want to get we want to get five to the 38 congruent to four mod 11 so Here's what I'd like what I'd like to have happen is if I just multiply this congruence through by four I like this three times four to go to one then I get exactly what I want But it does go to one because it's exactly one bigger than 11 Now we're done Okay, so this I'm gonna circle this up here and just keep in mind where everything's modulo 11, right? This is equal to one mod 11 Okay So then five to the 38 now is congruent to four mod 11. Okay, so here's what I'm gonna do Here's the last thing I'm gonna do Okay, and again, this is just for the sake of time. You're certainly welcome to use this fact I think I did I guess I forgot my book Everybody have this down The book kind of lists this as a lemma, so I'm just gonna call this a lemma Suppose Let's see. Actually, I don't want to I don't want to get this Wrong here. Actually, do I have it written? I'm gonna say yes. Okay Yeah, yeah, I've got it I still got it here. I'm just not gonna write the proof out suppose that P and Q are distinct primes a to the P Congruent to a mod Q and A to the Q Congruent to a mod P then a to the P Q is congruent to a Mod PQ and the proof is actually really easy really, but I'm not gonna do it. So What I'm gonna do instead to finish the day You may actually finish a little early today Is to do an example first before I do that? Let me before I forget. Let me give you homework from this section Okay, so this is 5-2. We'll have to talk about of course when this is due So Let's see So we have two two more sections probably you wouldn't like to have them both do at the final because then you're not gonna Get any feedback really on that We've got okay, here's what we're gonna do so we still have this week and next week All right for the final and Because I'll be talking about a bunch more problems on Thursday this assignment will be do a week from today one week. Okay, okay, so one two four six seven ten Okay, so now what I'm gonna do Start off with This is the last part of the lecture for today. We'll do number 2b Okay, now some of them do have you know quite a few parts, but once you get the hang of this you're not doing a lot of proofs You're just kind of You know modifying congruences until things come out. It's it's not really that bad. Okay, I'll show you what I mean with this example so to be And it says And most of these just boiled down to from Oz last year and they really do so if the GCD of A and 35 is equal to one show that I think this is 2b. Oh Is that what to be? Oh, it's the two two a is the GCD of A and 35 being one show that a to the 12th can grow into one month 35 Okay, sorry, it's two a then yeah to a okay, so Here's what's gonna do Keep them in mind some of your old tricks, okay, and then these will help you Going first of all I want to point this out, right? You can't directly go to from Oz last theorem you want to prove this, right? But from Oz last theorem has to do with congruences mod p, right? This is not a prime. Okay, you definitely shouldn't be thinking I apply from Oz last theorem to this You can't do it But what you can do is break 35 up into the product of two of two primes, right? Seven times five and then you can start thinking about from Oz last and you put everything together at the end So I will write that first just to make sure that you're aware of this note that 35 is definitely not So what we're gonna do instead is Well, we're gonna show two things Okay, so the first Is that eight of it and I'll if you're wondering where I'm getting this I'll explain it to you all in the end eight of the 12th is congruent to one mod Seven okay, that's the first thing we're gonna show You'll see in a second and then okay, so what I'm gonna do is I'm just kind of go through this And I'm gonna try everything in the at the very end and then if you still have questions Then I'll try to address those I think once you see how it all ties together. I think it'll make sense at the very end Okay, well, okay now think about this If we were to apply for miles last theorem here, there's this okay, so this is prime Right, so if we're gonna apply for miles last theorem It's gonna say if the other condition is met all we can say is then that a to the seven minus one in other words A to the sixth this congruent to one mod seven doesn't say anything about 12, right? So but there's still something we need to know this is important We in order to even apply the theorem we need we need to know that seven does not divide a that's one of the conditions on the theorem remember Right a to the p minus one is congruent one mod p when p does not divide a so we need for seven to not divide a We have to have that right? Yes Seven does not divide a okay, so otherwise the GCD of A and 35 I'm trying to get all this in here would would certainly be at least seven right you see that Seven divided a do also divides 35 so the GCD has got to be at least that that big But it's not it's one it's one so that can't happen Seven no So yeah, so if seven so if the GCD were not one so seven divides We don't know anything about a really so the only thing we can say is if seven actually did divide a Then seven would divide divide a and 35 and so then yes seven would have to divide the GCD It's the GCD would have to be a multiple of seven. Yes That's right. So By For ma's little theorem, right a to the seven minus one is congruent to one mod Seven right that's exactly what For modest little theorem says okay, we can we check the condition seven's prime seven does not divide a so we get this for sure and So that says that a to the sixth Is congruent to one mod seven right and now this is also a theorem from 4.2 now We can square both sides okay to get a to the 12 It's congruent to one mod seven okay, so that took care of the first one right any questions Okay, so what do we want to do for the second one? Well, it's similar right now We want to show that a to the 12th is congruent to one mod five again note that um Five does not divide a right otherwise the GCD of And 35 would have to be bigger than or equal to five right five divided a Then five divides both a and 35 It's common divisor, so the GCD has to be at least that the greatest common divisor of course has to be at least that Okay So then it's the same thing for ma's little theorem then implies that What does it say it says that a to the? Five minus one is congruent to one mod five right that's what it says right? In other words a to the fourth is congruent to one mod five and Now you can probably see where we need to go right? Just cube both sides right cube both sides I'm running out of room here, but we want to write that your notes. I'm cubing both sides Just like I squared both sides before okay, so this gives me a to the 12th is congruent to one cube which is one mod Five here's the punch line then now we're now we're done okay thus a to the 12th is congruent to one mod 35 this is something that We talked about actually before This is actually one of your homework problems from the last section remember there's a problem That's that something like you know solve 217s congruent to something mod 300 and whatever it was and the problem said break it down into these congruent You know into these separate congruences The whole point is remember that if your modular relatively prime if you have the same congruence But the modular are relatively prime that those congruences are the same then you can just multiply all the moduli together Into one congruence. I'm just going to say one thing out last thing about that before we finish Think about I just because I want you to I want to remind you of this right? What does this mean? I want you to think about this this will help you if you understand it It'll it'll help you make sense out of it later. What does this mean? This means that seven divides a squared. Sorry. This means that seven divides a to the 12th minus one What is this mean? This means that five divides a to the 12th minus one But since seven and five are relatively prime their product divides it that goes way back to a divisibility theorem for a long time ago So these are the kinds of things that you're doing you're just playing numbers games again really Okay, so I've done a couple of examples for you. I've basically done the whole section. We only have one more to do So we should have plenty of time to do that and review and do more homework problems So you really should be in decent shape for the final Thursday. I'll start trying to tell you more about what to expect on the final exam Okay, and just FYI the final exam is is May 16th So it's Thursday of finals week. So you still have a plane of time to study for the exam I believe please don't quote me until I verify this I think the the finals from 140 to 410 May 16th in this room. I'm almost positive. That's right You know, it'll basically be cumulative, but I'm gonna there's some things I'm gonna say about that But I'll start talking about that on Thursday So and if I had to say is it cumulative or is it absolutely not cumulative the answer is it is cumulative Okay, but you will be given a break in a sense, but we'll we'll talk about that on Thursday. Yes Mm-hmm. Okay this one


Hello from Hollywood. This is C.P. McGregor speaking, and welcoming you to another performance of Proudly We Hail, a program presented by your War Department. Through the courtesy of the Hollywood Coordinating Committee, we give you Mr. Randolph Scott as the star of our show to Desk a Highway, written by Richard Hall with music by Eddie Skravanik. For Chief Engineer Bill McElvane, building a mountain road in Peru had his headaches. For one headache, the fact that the replacement engineer requested weeks ago from the home office the JJ Dome Construction Company hadn't arrived. McElvane is fuming about this problem the morning our story begins. Jerry, where the devil are you Jerry? Coming, Mac. What are you doing out there, taking a siesta? Where's the mail? It's right here. The runner just got up from the desk with it. Oh, he did, huh? We put in ten miles of the best highway ever built in this country from that lousy town down there that runner is always late with the mail.  Yes, Mac. All right, now what have we got? Well, not much important except this telegram. Well, that's plenty important. Let me have it. Maybe they're sending me that engineer I've been desperate for this month after all. McElvane to Desk a Road Project. Last week's progress report very unsatisfactory. Please explain, sign JJ Dome. Jerry, this is it. Well, take it easy, Mac. Take it easy, nothing. I break my back. I sweat my heart out. I beg for an engineer and what do I get? Last week's progress unsatisfactory. Mac, there's something I... Jerry, I want you to do this. Yes, Mac. Save every one of those telegrams. Yes, Mac. When we get through here, I'm going back to the states. I'm gonna say I never had the pleasure of meeting you, Mr. JJ Dome, but right now I'm gonna have the pleasure of taking all those telegrams you sent me and cramming them right down your throat. You all through, Mac?  Blister him. Tell him the facts of life down here and ask him where in the devil that engineer is, he promised me. Yes, Mac, are you all through now? Yeah. Good, then I guess I can tell you. A new engineer arrived this morning. What? Why doesn't somebody tell me these things around here? Who is he? Stephen Grant's his name. Brother, how we need him? What's his experience? Who's he worked for? Nobody. He's fresh out of college. Fresh out of college? I knew it was too good to be true. I asked for an experienced engineer and they sent me a pipsqueak fresh out of school. All right, Jerry, get him down here. We'll use him somewhere. Grant, you're late. I sent for you an hour ago. I was shaving, Mr. McElven. Oh, you don't shave around here, Grant. We're too busy. Good. So you were just out of college, huh? That's right. And now you're going to be a great big construction engineer. Not going to be, Mr. McElven. I like to feel that I already am. I see. Well, tell me, where do you think you'd fit in best? That's up to you, Mr. McElven. You think so? Sure thing. Just something with a little responsibility, that's all. I like to think I can handle just about anything you want to throw my way. Oh, you do. All right, Grant.  When you get a few calluses on your hands, I'll give you a job with a little more responsibility. You're the boss, Mr. McElven. That's right. And don't you forget it. Well, Jerry, we're coming along. Lucky thing I got hold of that engineer down in Lima. If we had to use the product of Sunnys from the States, we'd be here 10 years. Oh, lay off the kid, Mac. He's doing fine. Oh, here he comes. Morning, Mr. McElven. Jerry, say, I've been wanting to talk to you, Mr. McElven. What you spoke about the day I got in? That'll come in time, Grant. Say, why aren't you wearing boots? What? You heard me, boots. Didn't anybody tell you about the snakes along this road? Well, yes, but... What do I have to do? Come over and dress you in the morning. You better get some boots on. See you later, boss.  Save it, Jerry.  Why? Why, he isn't even dry behind the ears. Say, Mr. McElven, can I have a minute? What is it, Grant? Well, I've been down on that grading crew for a month now. So what? I worked as a grader for five years and no complaints? Well, I'm not complaining, Mr. McElven. It's just that I signed on with this company as an engineer, sir. What are you kicking about? You're getting paid. You've just got to learn the practical side. I've had just about enough of the practical side. Now, listen, Junior, as long as you're working for this company, you're working for me, and you'll do what I tell you to do. McElven, there's a limit to what a fellow stands. Is there, fellow? Why, you? Look out, kid. Jerry! Jerry! If I get a cold water and douse him, when he comes to, he may want to get paid off. If he doesn't, keep him on the grading crew until he learns some manners. That's what I like about you, Mac. You're so considerate. In just a moment, we return to our play starring Randolph Scott. But first, a message from your ward apartment. If on a quiz program, the question were asked, what is the greatest team of technicians and scientists in the history of the world? The answer would be very simple. The U.S. Army. Yes, the Army is a group of trained and educated men continually working toward scientific achievement engaged in research and development here to fore unheard of. Every member of the Army is instructed in a particular trade or skill, contributing to the overall pool of knowledge and accomplishment. The recent discoveries by the Army in the fields of atomic energy, supersonic flight, push-button control of airplanes and new types of improved communication are but the vanguard of a new era of scientific living and improvement. But what are the personal inducements for a man to associate himself with such an organization? A private, just starting in the Army, has a pay of $75 in addition to free lodging, food, clothing and medical care. In addition, there are family allowances, extra pay for overseas service, flight or glider pay, longevity and unmatched retirement benefits. See your local Army recruiting station today about a career in the regular Army. The Tadesca Highway starring Randolph Scott is Bill McElvane. Trouble hit the Tadesca Road project to a degree which made J.J. Dones telegrams pale by comparison. First the cloud burst, which stalled the heavy equipment. Then a giant slide, isolating the construction camp from the town of Tadesca. Bill McElvane was not in his best mood as he discussed the situation with Jerry, his man Friday. Well, it looks like we're bogged down but good. That's right.  No, I wouldn't care to play a little honeymoon bridge. What happened to the kid? Well, Grant, as far as I know, he's sticking. I put him back with the graders. Well, who knows? Maybe he'll work out after all. Now, what about that load of TNT from Tadesca? Well, it didn't get through and we need it. All we've got is a few cases of that crystallized stuff. Is that still around? Why in the devil hasn't something been done about that? Well, maybe because it's liable to go off if you breathe on it. Well, you've got to get rid of it. Well, at least we've had a full dose. Nothing else could possibly happen. You think so? What's up? Look out the window. They're carrying somebody in. Looks like somebody's been hurt. Hey, it's the kid. Good man. Clam on the table. We've got to work fast. Rip the pants leg off. Now get the knife over there, Jerry, and the medical kit. Okay, here you are, Mac. First attorney kit. Above and below. Crazy kid. I told him about the snakes. I told him to wear boots. The knife, Jerry. Right here. There. That ought to open it. Well, that's all we can do here. Here. Well, we're not going anywhere. We're going to town. We've got to get into the hospital in Tadesca. Well, we'll never get through to Tadesca, Mac. There's a landslide. Or didn't you know? I'm going to blast our hole right through it. Get me the keys to that old truck. What are you going to do, Mac? Put Grant in a car. Give me 10 minutes start and then follow me. I'm going to get rid of that crystallized TNT. Mac, you're crazy. You'll never make it. Now hit the bullseye, babe, and don't miss. Jerry, the gate's open. Yeah, Mac. Let's get Junior to the hospital in Tadesca. He's been in the job in a few days. He ought to be. He's been sitting in that hospital for two weeks. I'd have been out of there in a day. Yeah, but you're not human, Mac. Oh, yes. A runner just brought up a wire from Tadesca. I'm holding him here in case you want to answer it. Let me have it. Michael Vane, Tadesca Bridge. Jumping Jupiter, Jerry.  I just received word of what you've done for Stephen Grant. Stephen insisted on taking his first job entirely on his own without any push from me. However, to realize Mrs. Donne's gratefulness and mine, I think you should know that Stephen is our nephew, whom we have raised as a son. Greatfully yours, J.J. Donne. Say, that's really something. How do you like that kid? Why, he didn't even tell me.  You want to answer this wire? Yeah. Send this to the old man. Save the hearts and flowers. What I did for Stephen Grant, I'd do for the lowest-day laborer in the section gang. Let's see. Oh, yes, you might add this. Leave Grant with me a couple of years. I might make an engineer out of him. This is C.P. McGregor speaking. I hope you've enjoyed our proudly-we-hailed story starring Randolph Scott. Before leaving you, Don Forbes has an important message for all of us. The young high school graduate today, has many professions from which to choose. One of the best of them is the Army. Our Army needs ambitious, intelligent men willing to learn the latest scientific developments into master-at-trade or technical skill. Trained men are needed to carry on its continuous scientific development necessary in this age of atomic power, supersonic, television, and radar. Army technicians are discovering and working on developments that scientists 25 years ago never dreamed possible today. The record of the Army's advancement in the fields of science is a long and commendable one, and each day is being materially added to. Such things as a parachute which enables a flyer to bail out at supersonic speeds, night-sight viewers, and planes which can fly faster than the speed of sound, represent only the beginning of amazing new scientific developments to be accomplished by the Army. As a member of such a progress-minded team, the American soldier is a man specifically educated in any one of innumerable branches of knowledge, such as electronics, atomics, medicine, industry, and aviation, to mention but a few. He is a technician. But besides the obvious and invaluable advantages of emerging a skilled man in his field, what are the personal benefits of a career in the new Army? Stated simply, a soldier is being paid for learning. A private receives $75 in addition to free food, lodging, clothing, and medical care. Not bad for a man just starting out in his profession, is it? After only 20 years of service, a soldier in the new regular Army can retire on a half-pay, or up to 30 years on a monthly retirement income of $185.63 for a first-term master sergeant. In an ordinary insurance retirement plan, it would cost approximately $1,000 a year to be assured of an income such as this after 30 years, but in the Army it cost nothing. There are many more benefits of enlistment in our new technically-trained Army. All young men between 17 and 34 should go to their local Army recruiting station today and find out about the opportunities for a career in the new regular Army. Thank you Randolph Scott for appearing on this program. Proudly, we Hale will come to you again over this station next week. Lesson in.