Although Euler began by disproving one of Fermat's conjectures, his later work moved Fermat's lesser theorem to a key place in number theory. In 1736, Euler proved a restricted form, and while most of us can read Latin, the type may be hard to read, so here's what it actually says. Suppose P is prime and P does not divide A, then P divides A to P minus 1 minus 1. Euler's proof is by induction. The proof is so obscure that it's periodically rediscovered by mathematicians great and insignificant. In 1747, Euler published theorems regarding the divisors of numbers which prove Fermat's theorem a second time. The induction proof is a little cleaner, but the method is essentially the same. However, this second proof is noteworthy because Euler also gave insight into how he found the factorization of 2 to 2 to the fifth plus 1. From Fermat's theorem, we can conclude that if P does not divide A or B, it must divide A to the P minus 1 minus B to the P minus 1. Again, by assumption P does not divide A, so it must divide A to the P minus 1 minus 1, that's Fermat's theorem. Similarly, it must also divide B to the P minus 1 minus 1, so it must divide their difference. Euler then proved that the sum of squares cannot be divisible by primes of the form 4n minus 1 unless both A and B are so divisible. And that's a nicely algebraic proof. Suppose 4n minus 1 is prime and not a divisor of A or B, then by the preceding we know the difference of their 4n minus 2th powers is going to be divisible by 4n minus 1. Consequently, the sum cannot be divisible by 4n minus 1, since if we add the two expressions, we'd get 2A to the 4n minus 2, which would have to be divisible by 4n minus 1. Now, since 4n minus 2 is 2 times 2n minus 1 and 2n minus 1 is odd, then we can factor, and so no prime of the form 4n minus 1 can be a factor of A squared plus B squared. Consequently, the sum of squares can only be divisible by primes of the form 4n plus 1 and 2, which is an odd prime. Next, Euler proved that the sum of 4th powers could only be divisible by primes of the form 8n plus 1. Since 8 to the 4th plus B to the 4th is also the sum of squares, we know that its only prime factors are of the form 4k plus 1 or maybe 2. But these numbers are either 8n plus 1 or 8n minus 3 numbers. Now, if a prime has form 8n minus 3, then it's got to divide the difference, but not the sum, and as before, the latter has a factor of A to the 4th plus B to the 4th, so 8n minus 3 can't divide it. And similarly, A to the 8th plus B to the 8th can only be divisible by primes of the form 16n plus 1, and Lather rins repeat to give us a main result. The prime divisors must be of this form. So let's look at those Fermat primes again. Any prime divisor of this 5th Fermat number must be of the form 64k plus 1. And we can list the first few such numbers, but of these, only a couple are prime. So if this number does have a prime factor, it's got to be among a very short list, and in fact only four trial divisions are necessary to show that the number is composite.


Now, the federal government has been advised to promote synergy among security operatives in the country and fund them adequately if the fight against banditry must be won. A traditional ruler from Ayyedireh, local government of Oshun state, or by Adhikolli Makama, the law of Kuta, gave the advice while reacting to the current security threats bedeviling the country. The monarch re-echoed the call for holistic restructuring and true federalism. He said that any money released by President Mohammad Abu Hari led administration for the procurement of arms should be used judiciously. Hello, hope you enjoyed the news. Please do subscribe to our YouTube channel and don't forget to hit the notification button so you get notified about fresh news updates.


What are the things that actually may be helpful in some cases and not necessarily helpful in others? Every situation is unique, both from the underlying problems that the individual has, that has the diagnosis, as well as the cancer type. And then finally, what treatment you're getting? One is mushrooms and mushroom supplements. So turkey tail, for instance, it's been used in Asia for decades as actually an adjunct to traditional treatments. And so there's actually a lot of data out there about it. But in terms of in more Western cultures, trying to incorporate it into our treatment regimens has been very difficult. And so I usually tell people, especially actually on immunotherapy, we are pulling back, I think in general on turkey tail use, it basically increases that immune function. And so we don't want to, you know, contraindicate what we're doing with immunotherapy by throwing in another immune stimulator. And I would say I don't necessarily recommend it with chemotherapy either because the FDA doesn't regulate supplements in the US. And so they're extremely cautious about brands and dosing and what's actually going to work and, you know, not cause a greater problem. And so a lot of those things, like turkey tail, I might use it in the survivorship setting where we're off of chemotherapy, we're off of immunotherapy, you know, trying to rebuild the immune system now, trying to recover. Same thing with something like lion's mane, another mushroom, it could be very good for memory and cognition. So when you have that chemo brain post-treatment, that might be another time to use that. So I definitely like to focus on what are things that are going to support the immune system as we were talking about the immune system is always on surveillance for these types of things and adequate protein in the diet or protein is a building blocks of ourselves and of antibodies getting adequate protein from the diet, which is especially challenging for those going through cancer treatment. But then we also have to look at getting adequate vitamins and minerals like vitamin C and zinc and vitamin D. And I think the problem is that people are looking for sometimes just this one, what's the one supplement I can take that's going to support my immune function. It's more complex for that. And we want to look at the dietary pattern that is supporting your immune system.


So here we have a solar car. So hello. Hi, I'm Winnie from the new solar team and we're here together on the ID Tech show with the rest of my team. And we brought our solar car Nuna 6, which is actually an older model. But we'd like to tell you something about our new model, Nuna 8. So you've been champion for the last 8 years, is that right? Well, the last 16 years actually. We became champion 8 times. 8 times in the last 16 years. And so who are you? I'm Pim Wume. I'm the mechanical engineer of the team and I've been doing some research for the past year. As we started September 2015 and we finished last October with the World Championships in South Africa. And who are you? I am David. I'm the electrical engineer of the team and as Pim said, we've been doing research for the last couple of years. And I was also driver of the car during the South Solar Challenge.  We drove actually on the public road. So our speed is just as normal as the regular cars. So we actually drove on a six-way, six lane highway on South Africa. It's really, really scared at the beginning, but then it was really fun. Scared? A little bit, a little bit. But I had a lot of trust in my teammates. In South Africa are the kids like running on the street or something? You have to avoid them or no? In some small villages you do have to do that, but mainly we drove on the highway. So that we have no problem. Did you drive over to the Kerger Park and look at the lions or where did you go? That would be really nice, but we did not do that. It was actually a race from Pretoria to Cape Town to see how much made the most kilometers. Pretoria to Cape Town and you won. We won, we made the most kilometers. So how is it possible, you're from Holland, right? How is it possible that you win so much? So we're from the Delta University of Technology and all of us got a really good technology-based background. So we got mechanical engineers, electrical engineers, civil engineers, aerospace engineers and so on. And you have more team here at the trade show, right? Who are the other guys and girls? Yeah, so we're with six people here and we also have our technical manager here. He's called Bo and he's making sure that we finish our planning sometimes.  We also have Crystal with us and she's our strategist. So she's been doing MATLAB all year long. So she can exactly calculate how much energy we still have left to finish the race. Did you follow exactly those calculations when you were driving? Yes. How did that happen? Did you have a list or did you radio? We have radio communication with Nuna, with the whole convoy. So that's really nice, it's really funny. It's like speed up, slow down all the time? Yes, we have a target speed and we have to stick to it. So that's in its various, the whole day according to the sun or the clouds or the environment and that kind of things. So let's look over here. So which model is this that we're looking at here? This is Nuna 6. This is version 6. This is number 6, we have built 8 cars during the last 50 years. This is Nuna 6. So this one won? This one unfortunately did not won, it became second in the World Solar Challenge. So that was like a wake-up call. Since then we have built two more cars and drove four races and we have all built it. So it was a really good wake-up call and since then we have been only winning actually. So what's different, this is the newest one? Yeah, this is the newest one. So what's the difference between this and this? There are two main differences, the races based on the regulations. The first big difference you might notice is that this is a three-wheeled car. You can maybe look down and you will see that there are only three wheels, whereas this one is four-wheeled drive car. Alright. The second difference is that this was the first car we built with silicon-based cells. So you see that they were cut, they were diced. That might be a really big mistake we did. So since then we have been driving with whole cells. Whole cells? Yes. So you are in charge of doing the solar panel here, is that what you were doing? During my project I was in charge of the new solar array we made together with DSM. We had a new laminate, we laminate the cells together and it proved to be really nice. Laminate together and is this one, people are saying is it true that this is the best solar panel in the world? Yes. The one that is on top of this car definitely is. That is the best solar panel? Yes. We made an improvement of 1% on top of the solar panel that we had already and that was already the best cells and the best laminates. So we did that together with our partner DSM and we have been doing research for the past four years already. Now for the first time we managed to place it on the car and race with it. So who is DSM? DSM is a really big Dutch chemical partner we have. They are also here at the firm and they also have a lot of, how do you say that, small departments. So they help us with everything from the carbon composites to build the car to the new technologies like the solar arrays. And so do you have to do a specific design for how you put them on the car and everything for optimal efficiency? Yes, from the beginning of the year we use a lot of calculations together with the strategist. Also some professors of the Delta University of Technology help us during the process. If we have questions we ask them and then together we try to make the most optimal car. So did you pass the year because you are students right? Yes. Now actually we stopped studying for the entire year so most of us just finished our bachelor's degree and then just got into this project for the entire year totally dedicated to building this car and do the optimal research. So in the last 16 years have there been 16 different teams or what? Actually no because it's a really good question and with a long answer. But usually you stay for one and a half year or something like that? Yes. So it's students? Students, all their students and for each new project we have a new team. New team. So that's it, you're finished now or what? Well, officially we finished in November but we are still involved in the team. A lot of questions from the new guys we get and also like this fair we're here. Are you saying you're not going to drive it again or what? You can just build another one and your next job right? How do you mean next job? I don't know what are you going to do in the future? What's your plans? I'm still not sure. I will firstly want to finish my master's degree in Delft and then I'll see. I think a lot of doors will be open with this project. Are there lots of people here that are trying to give you jobs? More or less. So you were doing the mechanical? Yes, and I did some aerodynamic research. So we saw at the beginning of the year the car couldn't be improved a little bit on the aerodynamic side. And that was also what the exam did with us. So the 3D printed prototypes for us which we could use in a wind tunnel so we could see if they were actually better. And it turned out they were. So we improved the car with that part with 2%. So that was exciting. So the kind of work that you were doing could be for airplanes, for cars, for everything? Well, the specific project that I did on the car was obviously for cars. I think most of the technologies used in the car you will see back at all sorts of technologies.  So I think on very, very much levels on the car you will see back at all sorts of parts in the communities. So right here you have a little table and you have a little corner here, the trade show.  Well it's actually more showing people what's the capabilities of the solar industry. So making people aware that it is actually possible to drive 60 miles an hour with the car just with 6 square meters of solar array. So that's exciting to see for people and hopefully opens their eyes. Is it kind of amazing to build this and then know that you're driving with your own work? And there's no gas or anything, it's just solar? That's really, really nice. I just need to think about it because you get used to it because it's a year-long work on the project, sleeping really less, making long hours. So you get used to it, but once you take a look back it was a really nice experience.  It's kind of cool, no? I think the best part maybe for you as well was the moment we started in Pretoria and we drove to Cape Town on the 5 lane highway that we was just talking about. And David was driving the car actually and I was in a lead vehicle and I think for me it was the most exciting moment because all the work came together really nicely and everything worked so smoothly. That was the best part of the entire year, just seeing your car driving on the highway with all sorts of cars passing by, that was awesome. Did you get like a stamp of approval from the South Africa or something? You need that right? You can't just put a car in the road. And they said no problem, just drive around, that's what you want. No, we need to, there are a set of rules and you need to fulfill those rules. So actually a person from the road authorities are going and they will check the cars, they will see if there's everything. All the nuts are tight enough and if you're able to support crashes, that kind of stuff. So this is the one you built right? Yes. So where is it now? Where is it? It just arrived on the Netherlands so that's why unfortunately we're not able to be here at the fair. And it's also nice for a new team to take a look at the latest car so they can know how it is. So what else is amazing on this one? What do you have? All kinds of technologies right? Yes. So what can you say? Like what's the motor? Like I told you, one of the biggest features was the Soda Array, but it's yours specifically for the motor. So it was actually located in the wheel and one wheel so we have an asymmetrical vehicle which is one wheel. Yes. It's also in the rim itself so it's a direct drive. It's provided by our partner Mitsuba from Japan and we also have a really good contact with them. So we have a custom-made motor from them. Nice. Every time you have all these companies you try to work with, you have to check if they want to support your team or the other team, right? Is it hard to work with Japanese because it's a Japanese team, right? Well actually the company I was just talking about was Mitsuba and it's a Japanese company. And they are very interesting in working with us. But we're obviously here, you're right, to just look for new technologies as well that people might be even capable of using on the next car. Is there rules that you can use, any technology you want? Is there any limitations? Yeah, if you can give you an example. For example, the Soda Array is based on what materials you can use. During our last race we were limited to six square meters of silicon or three meters of gallium arsenide. During the next race we are limited to four square meters of silicon or two square meters of gallium arsenide. Less and less, every time? Yes, because it's getting faster and faster so we need to reduce. So it's really nice to see that from the beginning of the project in 2001 we've been able to drive the same speed with less power. So that's really nice to see if you look back at our history. So soon the car will be like tiny and it will be fast enough to... So do you think this should go on the Tesla? Well maybe in a few years. Right now you cannot drive this on the normal road because you don't even have any mirrors on it or everything. But you never know what technology brings. So right now, no, but I think in a few years you might. And maybe it doesn't recharge the whole Tesla, but just maybe it adds five or ten percent to it. Yes, that would also be very interesting. They should just do it. Have you spoken with Elon Musk? We haven't met him yet, but it would be really nice if he... He's coming to the conference, no? Really? I don't know, maybe. I think his headquarters is just next door. Yes, but I have my business card ready so let it come. So you should go with this car.  You have a honk? That is a really good idea. And they all come out and like...  Yeah, they'll give you each a Tesla, but you have to leave the car. I think we can do a race with Tesla, maybe. How's the acceleration speed? Not very impressive, but I think a Tesla needs to charge up for about... What is it, two hours? Every 500 kilometers and we don't. So it might cancel each other out then. You only have 20 kilogram battery? Yes. And you can still drive 400 kilometers with one single charge? 400 kilometers with 20 kilometers without charging. That means you can drive at night? Yes, that's true. Or when the start starts to go down it starts to be not so efficient, but do you still drive during the night or is it usually it's finished? It's possible to drive during the night, but we don't do that again because of regulations. We just drive during the day. You don't have lights on it? We do have lights also. But it's not safe to drive during the night? How the brakes? Then he can talk a little bit more about the brakes. Well, also again the regulations just say that you have to be able to stop within 20 meters when you're driving 60 kilometers an hour. So it can exactly do that because that's the most efficient for us and we can just design the entire car around the regulations so that it is the lightest car, the fastest car. Everything must be the best and that's why... Are you working on the next one or the next team is working on the next one? The next team is working on the next one. So they already started in last September, but of course we're still very much in contact with them. Last month they started only? They started last September. One year ago? No, no, no, this September. So how soon they will have it finished? The water race is end of October. I think they will be starting the 8th of October around that date and then they will finish the car, obviously. So they have one year and something to finish? Yes. And get it to where? Where is it going to be the race? The next race will be in Australia. Australia? Yeah. So how do you get a car to Australia? You have to put it in a boat? And the boat takes like a month or something to get there? On the plane. We usually go with the plane. Oh, okay, on the plane? Yes. So it's extra luggage. You have to pay a... You can try but I don't think you will get it through customs. But you put it in a big container and then we ship it to Australia. How much it costs to ship? I don't really know. It's not your department, right? We probably wouldn't be able to tell you. She takes care of that. And we have guys in the team that take care of the logistics. Well, we get it there and then we drive from Darwin to Adelaide. That's a 3,000 kilometer race. So I guess the prime minister of Holland is pretty happy with you, right? Or what they called? I think he might be. He might be? He didn't say anything? Well, sometimes we get complaints of him. You get what? Just a small word of him. You don't get invited to the castle or something? No, we don't get invited, no. That hasn't happened yet. All right, because this is the pride of Holland, right? One of them. One of them, okay. There's some other things, like football and stuff. No, not anymore. Not anymore, okay.