The following is a presentation of TFNN, The Trader's Edge with Steve Rhodes, toll-free at 1-877-927-6648 or internationally at<PHONE_NUMBER>. The Trader's Edge, now Steve Rhodes. Good afternoon folks, welcome to the July 12th, the terrific Tuesday edition of today's Trader's Edge show. I'm your host, Steve Perseverance Rhodes, who absolutely knows that each of us should always be pioneers of our future versus prisoners of our past. Hope everyone out there is having a great day. Hey, let's make sure we have an extraordinary one and the easiest way to do that is to always remember that life is happening for us, not to us. That's right. So one little two-by-four shift means we can find the gift in every set of circumstance that life is going to toss at us. Now today you and I, we're going to go check on the circumstances of these markets. We're going to go figure out what those bulls and bears, what those buyers and sellers are communicating to you and I at just past one o'clock in the afternoon. I do want you to know I am absolutely grateful for your presence here, but more important than that, and that's this, during this next 53 minutes I'm here to serve you. So feel free to pick up that phone. You can dial on at<PHONE_NUMBER> if you can't dial in, we've got you covered there too. You can always send me an email, send it to Steve at TFNN.com and inside that subject heading, if you'd be kind enough to put, radio show question. Of course, in our Tiger's Den, well, any and every ping will do. So let's go ahead and get this show started on terrific Tuesday. Of course, this is Tiger, Financial News Network. I'm Steve Rhodes. Welcome to the show. Right now we've got a bit of a mixed bag out here, that mix coming from the S&P and NASDAQ, which we've gone just slightly negative, basically flat in the S&P, NASDAQ is flat as well.  She's trading at 6.15 to 30 year treasury, up at the 139.13 level, that's up by 22 ticks as we speak, lead to charge, dollar-wise the upside, you've got booking holdings, that's up 27 bucks, 1.5% charter, communications, 19 bucks, four and a quarter, Boeing is up 11 bucks, 8%, big move there, BlackRock, up 8 bucks, 1.3, 10th percent, SVB Financial, 9 buck runes, it's up a little over 2%. To the downside service now, up 53 bucks, about 11%, Paycom software down 24, 8% Thermo, Fisher down 19 bucks, 3.5%, Atlassian Corporation down 8 bucks, I'm 8% at 17 buck runes and Paloal Networks off $16, 3% to the downside. So what do we got out here? Good question, I tell you what, we've got a quick request, I was gonna go to it any by the G-Man, and that was a take look at Apple. The reason we're gonna take a look at Apple, go over into this three time frame, then we'll go into the actual charts, the equity future charts out here. Well, we can see about Apple, so there's an A to B equal CD to the upside, so we'll draw that in, and prices attain that level. Does not mean that the move higher is over, but the A point down here is January, sorry, June the 16th. The B point out here is to move up into the high on June 27th, and the C point is a move on June 30th. Man, put that in there. There we go. So you can see 148.22 was the one to one price projection for the A to B equal CD pattern. Well, the actual high today, 148.45, you gotta love it. Does that mean that the move is over? No, in order to complete a sell the D point pattern, we would need to see some type of bearish reversal candle. Don't think that'll happen today, but anything is possible. The other thing though, that we want to watch in the case of Apple, really a couple of different things. Let me just expand out the daily chart, make it a little bit easier. And the first is, or the second thing we want to take a look at, G-Man is the top of its daily profile. So we know where the snipers are sitting. The snipers sit at the top of a profile. That's where the sellers are at. That level, 147.55. We're trading 147.59 right now. And so we're up against a level of resistance. Now, if price can close above 147.55, that would be a bullish outcome. However, let's go take a look at this. We can also see that Apple is trading into a swing point from the trading session of June 1st. Now that had a volume of 74 million shares. If we just do straight line math, what I mean by straight line math, we take the 42 million shares that have traded so far today. We divide it by three and a half, because there's three and a half hours that have gone by. And we multiply that times 6.5. Well, that would give us a figure of about 78 million shares. So price is moving into, now I don't know what it's gonna look like at the end of the day. Obviously you've got the open to buy. So a lot of your volume isn't at the first 30 minutes, first minute, so to speak out here. If we take a look at what Apple is doing, if the volume holds up, it's moving into that June 1st swing point with volume. That had 74 million shares and we're maybe at about 78 million shares. Well, just, we won't know till day's end. But this is looking like, unless we get a bearish reversal candle that Apple wants to make more than a one to one A to B equal CD. That next price projection would be 152. So we really want to watch the top of that profile, 147.55. We want to watch that at today's end. If we take a look at the daily timeframe chart out here, the daily timeframe chart for Apple, my recollection is this is a TD9 count bottom that formed on June 13th. And so if price can clear on the daily basis, can clear the top of its profile, what that then does is that opens up the commona for price to move up to the 163, 167 level. We don't have that signal just yet. And price can need to clear that June swing point in order for that to take place. But likely what Apple is suggesting to us, at least from an intraday standpoint, is it wants to go spike the ball. Spike the ball says get up to the 151.74 level. And it'll especially give us that message if price closes above 147.68 or 147.60 right now. So that's what I'd be watching with regard to Apple. So how does that relate to the general markets? Well, let's go take a look at the charts for the NQ. So we'll switch panels out here and we'll take a look at the multi timeframe set of charts for the NQ. And the first thing that we will do is I've been to move something over there because things have changed just slightly I believe since we were last together. If we take a look at the NQ for its 60 minute timeframe, well what this chart is doing is looking at the, nope, it has not changed. It changed for a second and now it's back. What I was gonna say is, as the show was beginning 13 minutes ago or so, we had that little bullish crossover but that has given itself up for the 16 minute timeframe. Let's go take a look at the one hour. And what I mean by that folks is right now, as we speak at 1.13 in the afternoon, 36 instruments are trading above the top of their 60 minute profile, 43 trading below the bottom. In order for the bullish move to continue, it's certainly like to see this with a bullish crossover. The other timeframe where there's two or three other timeframes, you got the four hour timeframe chart. Four hour timeframe chart, so it's 37 instruments above the top, 33 below the bottom. That is bullish or a potential bullish signal. And if we take a look at the daily timeframe, daily timeframe still in bullish fashion, 27 instruments trading above the top of their daily profiles and 15 trading below. Of course on the weekly timeframe, that's where it's a real booger. And that is 63 instruments, oh, I'm sorry, eight instruments are trading above the top of the weekly, 30 below the bottom. So what's all that mean? That means that we'd likely have a choppy market, which is what we've had so far. No reason for that to continue if those market profile, market breadth levels change and they all switched with the exception of the weekly chart. If the other three, the daily's already bullish, 60 minute was slightly, no, the 60 minute was not, but the 240 was. All three of those get to their merry way. Then we should see a little bit more of a concerted rally out there. Here we've got the daily timeframe chart. What do we have today? Nothing other than taking out today's lows. The five hour chart formed a road spin indicator top and that pulled price right back to support. That was the bottom of the profile. But this morning's rally ran right into resistance. That's at that green, us that are in change line at 11, 985. Steve Rhodes with TFNN. We get back to this break. Let's go to our first request taking a Devon Energy and ticker symbol XBI, the biotech for the S&P 500, we'll be right back. Time of glooming inflation. We are purchasing powers eroded. There's no better place to protect your hard earned money than ain't gold. This the gold's flagship asset is the Monk Todd Gold Project in the Northern Territory of Australia. This is Australia's largest undeveloped gold project. We are talking a world-class gold project in a tail one mining district. This is a large scale, low cost project with significant existing infrastructure in a politically safe and friendly mining jurisdiction. This the gold just completed the Mount Todd feasibility study which resulted in a seven million ounce gold reserve in a 16 year mine life. All of this combined with the approvals of all major operational as well as environmental permits. This distinguishes Mount Todd as an attractive, devious pot ready development state gold project. This the gold trades on the New York Stock Exchange under the symbol VGZ. Steve Rhodes started his trading career as a student almost 20 years ago and the student has now become the master. Steve won the prestigious Timer of the Year Award in 2018 and barely missed that mark again in 2019, finishing at number two for the year, an amazing accomplishment. Steve Rhodes is committed to sharing his techniques and knowledge with anyone who wants to learn and he shares his vast amount of trading knowledge every day in his Mastering Probability newsletter. 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This is for Rich, Rich writes in and he says, happy two for Tuesday, could you please provide your analysis of DVN, looks to me like it's trying to form a bottom at the 200 day exponential moving average, so you're considering buying some. Okay, so you've got the 200 day exponential moving average, you've got that all figured out here. If we take a look at my chart, so you've got the daily, weekly and monthly that we're showing right now. The daily timeframe confirmed a roads meant to be indicator bottom pattern, it did it when it generated a three river morning star pattern, that was over the three day period of July 5th, July 6th and July the 7th out there. So support, you've got really two levels of support for Devin Energy, Rich, one is gonna be the bottom of his profile, 50.07, the second is gonna be the low of that candle formation and that's at the 48.86 area. Now, what Price did after forming that confirmed roads meant to be indicator top was it went right up to where it should have gone and that was that red oscillator and change line. And that's the issue here because Price has deflected that. That says we still have bearish momentum to the downside. In other words, we have a falling Price oscillator below zero. So this suggests that Price could pull back to 50.07. So I know you're looking for an entry point. So it's maybe, so you've got it figured out. I believe, let me just take a look at what the volume matrix is today as Price pulls back. So I'm gonna do this on my other panel out here. It's just gonna be a bit easier for me to do that but I'll share with you the results. So Price is pulling back into the July 6th swing point that had volume of 16 million shares. You're at 5.4 right now. So Price is pulling back with later volume. So if you wanna use your 200 day, I get it, that's fine, just know that it hasn't proven itself to you and a bit concerning that Price has been able to unable to take out that red oscillator and change line. If we look to the weekly timeframe for Devin Energy, we don't have a bottom pattern but we do have perhaps a bottom signal and that signal was that last week was a hammer candle. So we do know that Price is trying to hold that low. That's the low that we gave you that said if Price closed below that area, support will have failed. Well, the same thing on the weekly timeframe and then Rich, that would be signaling a move to 4131. Now the monthly timeframe chart shows 18.9 count top and it also shows that Price has pulled back and so far right now, Price is holding that green oscillator and change line and so for the momentum time period, what we have is really a neutral signal but support is held. So support is held on the monthly, a nice buy signal on the daily with caution because of Price rejecting that red oscillator and change line and the weekly says, well, I'm trying to form some type of bottom by giving you a signal of producing that hammer candle. So I hope that that helps you out. Let's take a look at your next request that was XBI. XBI, that is the ETF for the biotechs in the biotech sector of the S&P 500. It's my recollection. And you're looking, this has also XBI has been trending up for a few weeks, yes. Where would you consider to be a good entry point? So if we take a look at XBI, what this actually did was this formed a roadsman to indicator bottom back on May 12. Price makes a run, runs into resistance at the top of its profile. Looked like it was going to take it out and there were two sessions with a close above it but that third session said, cyanara baby, I'm going to go test support. Support was the bottom of that daily profile. Kind of similar to what Devin Energy is doing, although in the case of XBI, price was able to take out that red oscillator and change line. So really not the same. So I shouldn't have said it was similar at all. Similar from the standpoint of price pulling back into a roadsman to indicator bottom. On the XBI, I don't know if it was done with a lighter volume, but let me just check that out. So the swing point out there, which was on May the 12th, that volume of 25 million shares. June the 13th was a test was 16 million. The next day, 13 million. The next day, 14 million. The next day, 10 million. Then when it took off, which was that following trading session, it took off with volume, 26 million to the upside. That was your sign of strength out there for the XBI and that it's made that move higher. Now, what price has done, so today's move out here from a momentum standpoint, Rich, this was your bullish signal, your entry point out there because what price did, the oscillator change line changed from red to green. That's around July the 1st, July the 5th out there. And usually you get a topping signal when price pulls back and test that level. Usually, not all the time. In this case here, what took place yesterday was price got back, tested it and rejected that level. Now, there is a new profile on the weekly timeframe. That formed yesterday. And so your support level is 78, 59. So not only was price pulling back to test its green oscillator and change line, it was testing the bottom of that new profile, which is slightly bearish in structure. Slightly bearish because the center, Rich, is at the 82, 21 level and the top is at 84, 63. But for the most part, do you think at this stage here, I think that price will go target that 84, 63 level. I don't know that it's the best reward risk to get into it, but you were asking me, where is an entry point? You said a good entry point. You know, depends what your play is and for what period of time. We're in these shopping markets, so maybe short-term trades are really the thing to take a look at and do out here. On a weekly timeframe for XBI, certainly I can draw in an A to B equal CD pattern out there. That's the only bottom signal I would have. It doesn't really matter at this stage, whether it did form one of the bottom patterns that I utilize or not because price above the top of its profile. So that's bullish. The daily timeframe is bullish. And the monthly timeframe, what price did was it pulled all the way back to its third breakout level, it's 64, 30. And that area is held. And we can see that this has an oscillator and change line that also has changed colors. So with daily support holding, with the weekly breaking out above the top of its weekly profile out there, and with the oscillator and with the bottom holding a breakout level, more likely than not, what XBI wants to do over time is go target that 9831 level. And I'm not saying it's 9831 that's gonna hit exactly. That's what the number is right now for that oscillator and change line for its monthly timeframe. Woo! So I hope that helps you out, Rich, for your two for Tuesday out there. Thanks so much for taking the time to write in. We've got a request out here from Dr. P. Dr. P wants to take a look at, can we look at, oh, how about that? Looks like we've already done that. So say Steve, would you take a look at XBI? Okay, we've done that. Thought of biting around 60. So 60, what would that take us back to? 60, even if price could close below the 7850 level, and that's what you would need to see Dr. P. Then I'd be looking at first 75.94 and then maybe 68.86. But we don't have a message right now that suggests that that is like the outcome. Then you go on to say looking at it as long-term investment, bigger balance coming or back to a 60. So at this stage here, I don't have, I don't show anything, get with it, getting back to 60. In order to do that, you first gotta clear and close below 75.94, and that's not the case. You're looking at this as a long-term trade. I think what we have to do is get to a long-term bottom in the market out there. And I am unconvinced at this stage here that we've seen the actual low in the market out there. So from a long-term standpoint, you might be waiting until, so we know that the markets, generally speaking, have a couple of time periods where it makes significant bottoms. Not always, but time periods. Those time periods are the end of January, and really the end of, and the middle of October. Those are the two primary time periods where markets make bottoms out there. So the question is, what happens as we move into this October? That might be the first time, Dr. P, that you wanna take a look at a longer-term trade in this market. Steve Rhodes with TFNN, hope you rate that. If you wanna take advantage of this sector, now is the time to subscribe to my Gold Report. The Gold Report is a comprehensive look at the metal sector, as well as the markets that move gold, which is the currency and bond markets. New subscribers get a 30-day money-back guarantee so you have nothing to lose. Every Monday morning, I publish the Gold Report with coverage of gold, silver, bonds, the XAU, HUI, GDX, as well as more than 30 different mining equities. To sift yourself the types of profitable trades that are recommended within the Gold Report, sign up now by visiting TFNN.com. Don't miss out on the next great gold trade. Sign up today. TFNN is excited about our new software charting program, The Art of Timing the Trade Charts. 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For more information, just click the Think or Swim banner on the front page of TFNN.com. Welcome back, folks. What I've got up on my screen here is the 125-year history for the Dow. And what this shows is, so I mentioned before we go into break, typically you see significant, the significant bottoms typically form at the end of January. So here where my cursor's at, that's around January 21st on average. Then we take a look at another significant bottom that typically forms. That's the one that's in October. Here, it's for the 125-year period it's got at about the October 26 timeframe out there. Now, let's go take a look at, let's go back and take a look at XBI. Since we had two people that wrote in about it, give me a moment here too, why isn't there XBI? Go ahead and pull this up here. And it's got different patterns. Now, if we take a look at XBI, which we know is in a bullish run out there, we took a look at it. In fact, I think we came back and said, yeah, likely we should, should see price move up towards that 98-34 level. Well, in the case of the XBI, traditionally speaking, or at least while let me take this back, 16 years. Yeah, so over the last 16 years, XBI has formed its top in the third week of July. That's taken a lower into August. Then you have a nice rally into the middle of September and then pull back into October, October 27. Now, that's a traditional pattern. Only have 16 years worth of data. So, does it really make sense to put in the midterm election cycle process that gives us about five years out there? Well, we can put it in. And so we're looking at five years worth of data. And this is suggesting in the case of XBI that it makes a bottom around July 17th. But this is not the pattern that I see that's in play as we speak right now. Instead, it looked like it was more of the pattern that we just took a look at when we took a look at all years out there. Let's take a look at Devin energy. So let's take a look at DVN out here. And whoops, I started at DVN. There we go. See what it is doing. And so with regard to Devin energy, we've got 36 years worth of data. Now, in the case of Devin energy, ticker symbol there was DVN. We know that that was pulling back. And that was pulling, even though it had a daily, I believe it was a daily rogment to indicator bottom signal, price is pulling back, lighter volume out there. But I did say that in the case of Devin, price ran up, hit that red oscillator and change line and then rejected it and it's pulling back. So we still have that falling price oscillator blow zero. So it's possible that Devin energy, now typically in an annual seasonal cycle process, doesn't usually bottom until about the end of July and then it moves higher into the November timeframe out there. So that's the normal season. Now, because we have 36 years worth of data here, I feel more comfortable when we put up the midterm election seasonal cycle to say, okay, maybe this is what's happening. Now, in the case of Devin energy, this shows a top or significant top around the middle of May. The actual top that came in, significant top that came in was in June out there. So it wasn't in May, but nonetheless got that top and price has been trading lower ever since. And so in the case of Devin energy, boy, look at this, doesn't typically look that good, but that's the seasonal pattern during the midterm election cycle out there. So I just thought that I would share that with everybody in the audience. We've got a couple more questions that have come in. So let's go to those. The first one coming in from Hector and Patty. And Hector and Patty want to take a look at the energy sector, the XLE. So he says, Happy Taco Tuesday, thank you. XLE, when the shooting stars appeared at the beginning of June, we dumped our XLE wagons. Is the XLE by the D point back on? So great question. Let's go take a look at the energy sector, the XLE. And because he's talking A to B equal CD patterns or Patty and Hector, are we're going to go back and take a look at the black background screen and then we'll switch back over to the white charts out there. So the black background screen, what Hector has asked is, is there an A to B equal CD pattern? The answer is, if there is, let's see, the swing point that I would look at would be June 23rd, 56 million shares. When that level was passed, it was with 48 million shares on July the 6th. Today you're down with only 22 million shares, the price below that swing point. So do we have an A to B equal CD? Certainly not on a daily timeframe. It hasn't been confirmed. It could confirm. And if we did get a confirming A to B equal CD, then what you're looking at is not lining up those wagons until we see price get down to the 53 level or maybe 46 out there. Now that's coming from the daily timeframe. But that at least answers the question. We do not see an A to B equal CD pattern out there to help you load up the wagons. We are trading below the bottom of a bullet structure daily profile, below the bottom of a weekly profile. And that suggests that what Price wants to do is make its move to 6416, the bottom of the monthly timeframe profile. Now let's go back and take a look at the white background charts. They may have other signals for us that we need to consider. And while a lot when we take a look at those signals out there, what you have very much like Devin Energy out here, the XLE formed a rose momentum indicator pattern. It's did it a couple of times. Well, just one time. This three river morning star pattern out here. But again, in the case of the energy sector, Hector and Patty, what's Price do? Gets up to that red oscillator and change line and gets deflected. And now we're trading below the bottom of that profile and it's trading into its swing point from July 6. Now the July 6 swing point out there has volume of 48 million shares. So far today we've done 22. That's light in the loafers out there. But nonetheless, if you do close inside that swing point, you could go down and test those lows out there. So you got a valid bottom on the daily. If you can test that low, that low being 66.45, do a lesson 48 million shares by day then that could be your buy signal out here. And perhaps Price would get up to the 74.88 area. If we look at the weekly timeframe chart, no signal at all of the bottom. In fact, this says you could see a move to 54.26. In the case of the monthly chart out there, you've got 65.99 as your potential level of support. And that's that green oscillator and change line. So a monthly TD9 count pattern, Price likely to test at 65.99. And I say, do not back up those wagons and load up out there Hector and Patty. Just don't see it in the chart patterns. So I do want, now we can take the XL lead. Let's do where am I at screen wise. I've got to change it to this screen here.  So don't know what that is, but we've got this screen here that we can use. So what we'll do here is put up the XL lead or XLER, which is not a symbol, by the way, to my knowledge. And we take a look at the XL lead. And again, we're just looking at the seasonals out here. So from a seasonal standpoint over the last 23 years, the XL lead should continue to move lower into when? October. Yeah, so that's a potential. Of course, we really got to take a look at late-speed crew to see what it is doing out there. Now that's without take a look at the midterm election cycle. Well, if we take a look at the midterm election cycle, says the XL lead doesn't generate a bottom until typically the third week of July out there, specifically July 22nd. But we don't use these. We use these as guidelines, not as exact dates out there. So that's what's going on with the energy sector, Hector and Patty. I hope that helps you out. Thanks so much for writing in and have a terrific Tuesday. Terrific, Taco Tuesday specifically. Let's go to Yvonne, who is asking about the doctor, Dr. Copper and Freeport McMaran, looking for an entry point. So thank you, Yvonne, for that kind comment out there. And very, very, very nice of you to say. So let's take a look at Dr. Copper. So for the Copper contract, we want to go to my other charts out here. We want to go to this set of futures charts. Now, in the case of Copper, I'm thinking it's September. Let me just make a double check out there. Is it September or is it September that's a contract? No, hold on, hold on, hold on, give to Stevie. It is September. So now we've got the HG0922 out here. Let's go see what it is signaling to you and I. So give me a minute for this to populate. So on the daily timeframe out here, in the case of Copper, let's open this screen up. What do we have? All right, so we Copper generated a confirmed by the D point pattern. And it does it back here the day of July 6th when it generates a bullish hammer candle. Now price is trading into that hammer candle. And if price were to close below its low, that low out there would be $3.27. And that would suggest that price is gonna continue to move lower out there. That's the daily timeframe. So let's do this, Yvonne. When we get back from this break, we'll finish, take a look at the doctor. Try to give you our best assessment for what is communicating to us at this moment in time. We'll be right back. 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Yeah, folks, we're taking a look at the doctor. What I did hear, Yvonne, is I went ahead and put the weekly chart up on our screen for us. So both the weekly and the daily timeframe formed those bottoms out there and the weekly's got that nice hammer candle. And the price closed below that. That low, by the way, was $3.27. That then is going to signal a move back to the 307 area. Now you can see the A to B equal CD pattern out here on the weekly timeframe. And that was confirmed with that bullish hammer candle. If I didn't mention that, I think I did, but I certainly want to mention that. It closed below the low of a hammer candle. It says if you're wrong, you're wrong out there. So that's coming from the weekly timeframe chart for Dr. Copper. If we take a look at the other intraday time periods, like a five-hour timeframe chart, what we see here is that prices pushing lower, less relative energy. A bullish reversal candle would suggest for that timeframe, a bottom has formed. You're below on the 240-minute chart, you're below the following bar number nine. And if it closes below that area at two o'clock, and that's 15, 17 minutes from now, no reason to think that that won't happen out there. That's negating that pattern to suggest lower price. You are at bar number eight on the 120-minute timeframe chart. So we're getting conflicting messages out here. I'm just narrating what the charts are communicating to you. Sometimes we get that conflict out there, very much like the markets. When we took a look at market breadth, we took a look what the other messages are. It's really conflicting signals. All would suggest really chopping us out here. So I think with regard to the doctor, you need to hold off and wait at this stage and see how the next couple of days play out. But you've got your parameters at least the key level of a bottom to pay attention to. So I do hope that helps you out with regard to the doctor. You also wanted to take a look at Freeport-Macmaran. So I'm wondering if you were asked about Freeport-Macmaran because you were saying to yourself, U-D-U-S-D, that, hey, if copper's formed at the bottom, then maybe you want to take a look at a long position in Freeport-Macmaran. The real question or the real thing that you want to look at for Freeport-Macmaran is the Australian dollar, which I've just punched up on the screen here. And in a moment, I'll show you that that's the real correlation, or at least that's another correlation. And I consider that to be a significant correlation. So the question is, has the Australian-US dollar formed a bottom? Well, it turns out that its TD-9 count bottom was negated yesterday. Price closed below bar number nine out there. That low was at 0.6763. Closed below that yesterday. Right now we see a bit of a rally, but that rally should find resistance at the center of that, well, I take that back. Can't say that. I would watch the red oscillator and change line. That right now is at about 6790 out there, but I don't see a bottom inside the Australian dollar because of yesterday's price action out there. And why is that important? Well, the reason that's important here is we're going to put this chart up on the screen. And this is a chart that shows, it's a different correlation chart than what I normally show out here, but here just looking at the two line charts. So this is on the close. And you can see the directional correlation that exists between the two. I mean, it is as direct as you get out there. It's not exact. I didn't say exact, but I did say it's direct. So let's go take a look at the charts for Freeport and Mack Moran, just to see what kind of signal they're generating for us. So F's, what the heck happened there? Okay, so now let me get the FCX charts fired up here. And we're gonna go ahead and change our screens so we're no longer looking at the Australian dollar. And well, that was wrong.  Screens? Okay, here we go. So now we're taking like a Freeport and Mack Moran. And we've got the daily, weekly, and monthly timeframe charts. Let's start with the monthly chart. The monthly chart has wave number seven and a roads mint and indicator top. Prices below the bottom of its bullish structured monthly profile just crushed that last month. That is suggesting that Freeport and Mack Moran wants to move lower. Now the move lower, quite frankly, is saying $9 in a penny. I'm not gonna go that far and say that's where it's going. On the weekly timeframe chart, you can visually see the A to B equal CD pattern. I'll just draw in the A to B line out here. And then we can go put, just take that line and move that over to the C to D level out here. There's your C to D. So you can see that price is nearing that one to one level. That does not mean that's where price will stop. That's around 25 bucks. Could be a 1.272 expansion, 1.618 expansion. Could be just headed back to its breakout level 23.77. The weekly chart does not have a bottom. Because you've got that A to B equal CD pattern out there, Yvonne. If you're looking for a long position, I prefer to wait for a bullish reversal candle on the weekly timeframe chart. We can't see on the daily timeframe that there was a Roadsmen Dominicator signal that lasted for one day. That was on July 1st out there. Curious, what was going on on July 1st with regard to the Australian? No, I don't know the answer to that question. I truly don't. Let's just go take a look at July 1st. What was happening on July 1st? July 1st was what didn't trade. Oh, it did here. That was forming bar number nine of a TD9 count. That's the TD9 count pattern that failed. So we go take a look at, so on that same day, when you had a TD9 count on the Australian dollar, what you also had was a bottom signal, hammer candle confirming the Roadsmen Dominicator bottom, for the daily timeframe. But that pattern failed after day number one. His price closed below the bottom of that panel out there. And now you're waiting for a Roadsmen Dominicator bullish reversal candle to confirm that Roadsmen, the next potential Roadsmen Dominicator signal out there. Nonetheless, Yvonne, I don't know what your time period is that you're looking for an entry. I would be focused more on the weekly chart and I would also pay attention to that Australian dollar. So hope that helps you out. Thanks much for taking the time to write in. We got a request out here from San, oh, nope, the first one coming in from S&P wants to take a look at BG. So let's get BG out on our screens out here, see what they're communicating to us. And BG is, what? You think I'd have these memorized? And I do not. BG is bunch limited, not in just watching. Well, if you were looking to get in, it appears that today is going to be your confirmation of the Roadsmen Dominicator bottom. You already had wave number seven, that was letter G. That was confirmed on the trading day of July 7th. It was July 6th, that formed that seventh wave move. So now we've got two bottom signals out here. Price is trading, this is bunch limited. It's trading into a small little swing point from back on the trade. Well, you really could say there's two swing points. There's the one from just two days ago, July 7th. That had 1.9 million shares. You're doing about 800,000 shares of this peak right now. The other swing point that I was really looking at was June 28th, 1.2 million shares. Again, you're only at 792. So you're moving into that light and you've got resistance. So the resistance zone out here, SNP, is going to be between 90, 13 and 92, 24. But on a daily basis, you do have a confirmed bottom that may just be, and price above is read-off, certain change line. Nonetheless, it still may just be a counter trend move up to 92, 24. We just don't know. But you've got a slightly bearish structured profile. So your resistance zone, again, 90, 13 and 92, 24. The weekly timeframe does not have a bottom signal. You've got an A to B equal CD pattern. You can see that out there. Again, it's got a TD9 count top and a Roadsman to Minicator top. This is more than a one to one A to B equal CD. Here's the A to B level. And when we move that over to the C to D leg, you can see we're well beyond that. So you're in the expansion zone. What you're waiting for here is the bullish reversal candle. See how the one to one on the weekly, or really the daily two would have taken us to about 96, 89. This is the reason that I say you do not buy a D point just because you've done the one to one level out there. You need more than that. And the way that the market communicates to us, the buyers and sellers, they check in each morning at 9.30 before they walk on that floor, so to speak, at a forward leap. And what they do is they produce bullish or bearish reversal candles or sometimes just a candle that is neither a signal. When you complete a pattern, look for confirmation, bullish or bearish, head out of the direction that you're taking. G Broads with TM&M. Vista Gold owns and operates the largest undeveloped gold project in Australia, the Mount Todd Gold Project. Vista Gold just completed their feasibility study resulting in a 7 million ounce gold reserve. Vista Gold has all major permits approved and has retained CIBC Capital Market Assistance in evaluating alternatives and in completing an accretive transaction. Vista Gold trades on the NYSE American and TSX under the ticker symbol VGC. Vista Gold executing a strategy to create shareholder value. You might think that if you want to be successful at trading in the stock market, you're going to need a crystal ball. After all, it's impossible to predict the future, right? 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The Tiger's Den, available to all tigers and tygruses for just $1 for the year. There's no catch or at costs when you join our community of traders. Sign up today and become a part of this educational community of traders. Just visit the front page of tfnn.com. Don't forget, you can listen to Tfnn live on your mobile device 24 hours per day. Go to tfnn.com and hit watch Tiger TV. That's tfnn.com and hit watch Tiger TV. Welcome back folks. So just to summarize BG out there, even though you've got that bottoming pattern that's suggesting that caution and to weight on that. So that was a signal there. Dan was asking a question about the spot ball of Tudik, sorry for overlooking that Dan. If we take a look at it, what today we saw was price in essence, testing and rejecting this 50 day exponents moving average. Price didn't get all the way up there. We got up to a high of 27, 38, the 50 days at 27, 58. As long as price remains below that, that says be caution on shorting, not from an intraday standpoint, but from a longer term standpoint out there. And that's really all that I've got to say about the spot ball of Tudiks at this moment. The last question was to take a look at the GDX out here. We'll go switch back over to the daily time, whether daily, weekly and monthly charts for the GDX. We'll look at our three panel chart out there in the daily time frame. We've got a TD9 count bottom. That TD9 count bottom is being attacked as we speak. And so the question is, what's the volume on this pullback right now into that swing point? That swing point, which was from July the 6th, had volume of 24 million shares. You're pulling back with 10 million shares as we speak right now. Now, there is a bullet structure daily profile. So you'd like to see the GDX close back above the bottom of that level. The bottom of that level is 2641. If it doesn't close above it, you may go test and perhaps take out below from July 6th. That was your TD9 count bottom. No such bottom on a weekly time frame, no such bottom on a monthly time frame. So on the daily, you want to see the GDX close above its oscillator and change line. If it does that, that would suggest a move to 2866 and above that, 3098. Do we have a confirmed bottom out here?  It's really like to see some kind of solid bottom in the case of Goldilocks out there. And it does have a TD9 count bottom. But as I mentioned earlier, as we began the show or during the market update, what you need to see in gold that's got a hole that's TD9 count bottom, which is 1730-70 or 1724-50. Hey, folks, thanks so much for joining me here on Terrific Tuesday. Stay tuned. You've got two more wonderful hours up next. Your favorite polar bear is up next after that. Tom O'Brien will take us on home. I'll be back with you tomorrow. Wonderful Wednesday. One o'clock, sure. Let's get in.


So a couple of years ago, I came here to Tic-Tac, and I was describing a project that was in the field, and now I'm happy to get back to Tic-Tac and describe the findings of the study. Where's the clicker? Well, kind of funny, because the results are a bit disappointing. I still hope that even though the results or the findings might be disappointing to some, there's going to be important lessons for future work, and I hope also another goal that I have is that people will see the value of collaboration between practitioners and academics. This is joint work with Melina and Jonathan Rodin at Stanford. Is that not working? OK, so I'll just start with that. This is just like a motivation, like the project started because we were all observing this exponential growth in the penetration of mobile technologies. This is a figure from Africa. And we see these types of figures, we were asking ourselves, OK, so this is a serious revolution. Can we use mobile technology, this penetration of mobile technology, can we use it to improve service delivery in low-income countries? Most of my work is in sub-Saharan Africa. So just to preview what we're doing in the study before we can dive in, we collaborated with local NGO in Uganda, and together we were looking at the effect of a community reporting platform, and we were looking at its effect on service delivery outcome in health, education, and water. But the platform basically did it allowed citizens, villagers that live in really remote areas, to report service delivery problems by texting to district officials in the district capital. We introduced the platform. The study was run as an experiment. So we had 200 villages, 100 were selected to receive the platform, and 100 served as a control. We followed them for two years. And though we found that utilization of the platform was relatively high compared to benchmarks from other studies that tried to crowdsource information with mobile technology in low-income countries, we're not finding large effects on service delivery outcomes. In terms of the starting point, the reason that we kind of like, where's the need coming from? So the need comes from this persistent abysmal public service delivery in many low-income countries. About 30% of teachers and nurses are not present in any given moment. Even when they're present, there's low effort levels. A lot of teachers are sitting outside taking tea, and especially in the health sector, there's a lot of abuse by health providers. And on one hand, the government has the mandate to discipline frontline service providers. But there's a problem, and one of the problems is that many local governments don't have necessarily the kind of reliable targeted information when exactly a teacher doesn't show up, when a clinic is closed, when it's supposed to be open, or when there's a nurse that is abusive. And on the other hand, citizens, the villagers, do have this information, but they might be reluctant to take action because there might be a repercussion. And so that is the status quo that we are, that is the status quo before we started the project. And so there's some good reasons to think that the technology might help us solve some of these problems. And so here's some reasons why we thought we might see an effect of service delivery if we implement this platform. If we connect between villages to the local government with text messages, text messages are very inexpensive. We actually set the platform in a way that it will be free. So in our platform, text messages were basically free, so that reduces the cost of reporting. Probably the most important aspect of this is the anonymities. Citizens in low-income countries are afraid to report on teachers and nurses because of power disparities. And so we set the platform to be anonymous. So you can send a text message, but on the side of the government, it just opens as a case. With a case I did, there's never a phone number. It also immediate, I don't need to wait till I go to the district. I can send it immediately when somebody abuses me or when the clinic is closed. But it also can allow government, if they respond, I can see that the government is responsive. And this can have rippling effects on my willingness to continue engage with the government. And the last thing is these platforms have the take advantage of the comparative advantage of villages in terms of the information that they have. But we're not expecting them necessarily to hold service providers directly to account. They just are providers of information. The information is just crowdsourced from them. The public officials are those that need to take the disciplinary actions. And so we were expecting that if we implement such platform, we might see an effect on things like monitoring of health providers by the district officials, maybe more effort of service providers, and maybe also more input. So this is what we set up to find. In terms of our research design, we were working in one district in northern Uganda called the Rua, districts in Uganda are responsible to the provision of services like health and education and water. In terms of the implementation, this is a collaboration with a local NGO called GAP. We launched it in 2014, and the end was somewhere in 2016. As I said, we introduced it in 100 villages and 100 villages in control. Because the platform was open, the treatment here is really just to go to villages and introduce the platform and encourage them to use it. And our unit will be village clusters. And so this is kind of how we set it up. Each health clinic in Rua, there's services for five villages. And so all four or five villages around the health center receive the same treatment. Yellow will be treatment, and blue is the control. The treatment was quite extensive. It wasn't just the design of the platform. There was also training of district officials on the dashboard. So on the side of the district officials, there's a dashboard. They can log in, they can see messages coming in, they can respond, they can also do some queries like how many messages come in a month, in a day, in the last year, how many in water, how many in education, how many, so it's a pretty sophisticated dashboard. And we had to train, now when I say we, this is the NGO that we were working, they were training. There were inception meetings in these villages to introduce the platform. There were quarterly meetings in these clusters where government officials came to report, here's what came in, this is what we did. This is why we didn't do things maybe because we don't have a budget, or maybe because it's the responsibility of the national government. There was a registration campaign also in 91 villages where we registered the phone numbers and sent periodic kind of text messages to encourage people to use the platform. This is how the inception meeting looked like. And I'm gonna, the first results that I wanna show is a result about take-up. We worked in about 100 villages, so the project was in 100 villages, but they are really tiny villages in a very, very rural area in Northern Uganda, in West Nile, and so this is an area where the villages are around 200 people and only about a quarter have mobile phones, okay? And almost no one has a smartphone.  At the top you have the number of messages over in the first year. These are cumulative messages at the bottom. The figure is monthly. This is, sorry, the mean on a monthly basis, where the dark black relevant messages coming in and the gray actionable messages, okay? And so we were getting about something like at the peak, something like 200 to 150 messages a month from a really small, very small population. What is relevant and what is actionable? This is really important for our analysis of what happened. There were lots of pretty significant number of relevant messages, like I greet you all, but major problem is sickness. It's clearly relevant to service delivery, but there's not much that the district can do with such a message. Actionable messages like the one at the bottom where there's like, it's very clear exactly where the village is and exactly what is the problem, okay? And so kind of previewing what we're finding, there were just not enough actionable messages. We were looking at health. We were looking at schools and we were looking at water services. And these are the outcomes that we were looking at. Monitoring, effort and input. So for example, monitoring is by the district. We're looking at the calls that the district officials made, the visits that they made, reports that they wrote about these facilities. Efforts could be looking, we were looking at absenteeism, we were looking at engagement. So we were sending people to schools and we're seeing whether something is written on the board, whether the teacher is actually in the class or sitting out taking tea. And we have a big part of the data we were collecting was also in inputs like drug stock out and supplies. There's a lot of data sources that we utilized in the two years of the project. We were looking, we were doing unannounced audits in schools and clinics. We collected tons of administrative data in the district from the district capital. In order to understand a bit also the results we supplanted that with a service that we conducted in 16 villages with over 4,000 respondents. And we also conducted some focus groups in eight villages. In terms of the finding of the study, so a little bit disappointing, we found no evidence at all of any change in health services, not in monitoring, not in inputs, not in effort. We do find effects in the education sector in all these three parameters, actually quite significant and quite large or substantively, but only in the first year of the program. These results did not last beyond the first year. And so we're actually happy that we did it as a two year project because if we would have stopped in the first year we might have been reporting results that are inaccurate. We did find also suggestive evidence of positive effects in water services. There was a lot during the two years we visited the district many, many, many times. There was a lot of anecdotal evidence where people were saying, oh, we send the message about that and then they fixed it and the district was happy to share a lot of these types of stories. Outtake is that this is great but what's the counterfactual? These things might have been also addressed anyway in alternative means. And so for us, even though there's some suggestive evidence of effect in water and there were clearly effects in education in the first year, we don't think that this platform had like a sea change in terms of changing the equilibrium from a low equilibrium to some high equilibrium where teachers are fearing that people will report on them and so they all start coming to work. Okay, so why didn't we find, so we have a paper, it's online, I'm happy to share with people that are interested. We go deep, I won't be able to go that deep in this presentation but we go and try to understand why do we see in education and not in health so we go through a list of things that could account for that so they're up here. It might be that it's just what people might care about more on education than health. We show that that could not be the case through survey data and through like message data. It might be harder for citizens to identify problems in health. So there's a whole list of things. At the end of the day, a lot of the, it's for us it's just putting out different reasons, trying to find evidence to some and not others but we wanna remind also people, we remind people in the paper that our N in terms of the district is equal one, meaning we have only one district education office and one district health office and it might be very idiosyncratic, okay? And so these are some of the limitations of the research design. Why did the effect in education didn't last beyond the first year? It could be that the first year was like novelty effect, people were excited. It could be that in the first year, the mobilization efforts of the NGO were significant. They did a lot more work in the first year than the second year. They were doing other things in different districts but they're also things that are related to citizen engagement that are really important. So on one hand, citizens, there was like something like 84% response rate from the district, which is really high like and that was not our prior when we started given what we know about responsiveness of governments in Sub-Saharan Africa but when you go deep and this came up from focus group, from surveys and from analysis of all the responses. So we basically analyzed all the responses of the local government, here's what we found. We found that because the messages were relatively vague on the side of citizens, the responses of the district were not that helpful for citizens so they were disappointed. Many of the things that they cared about were not the district couldn't address in the budgetary cycle and a lot of time the responses of the district government were technical, not understanding the reality on the ground. So the classic example is someone says the teacher in my kid's class never shows up and the response of the local government was oh, okay, this is really important, you should talk to the principal. Kind of da, I mean the people know that they can talk to the principal but the whole idea of reporting anonymously is because they don't wanna confront the teacher so after they receive such a message they basically disengaged, okay? And so this can account for some of the problem. At the end of the day we found that this had an adverse effect on citizen satisfaction. So we did a survey kind of to also explore a bit some of the assumptions of when such a platform, we were thinking when such a platform should have an effect. So first, citizens need to believe that the district can fix a problem. If they don't believe they will never engage and so we find that this is the case here. Our survey suggested clearly people think that the district, if it wanted, has the capacity to make a change. Citizens need to also believe that the government will be responsive. Why will I send a message if I think that no one will be responsive to my message? And we find also, and that was also a positive finding for us that citizens did believe that they can prompt the government to take an action. Obviously in order to use the platform people need to know about it. So we found that only about 30% of people knew about it. I don't know, many of you can tell me if this is low and high, I don't know. But as I said, there was, you know, in order to have an effect, citizens must send actionable messages and this is not something that we found. I'm gonna finish with a few lessons learned both on the side of citizens, the side of government and the side of technology. On the side of citizens, based on the take up, based on focus groups and the surveys, I can say absolutely clear that there's a demand to use these types of platform. So it's not as if we try to solve a problem that doesn't exist. I don't think that this is the case. There's absolutely a need in poor areas for these types of platform. One of the things that came up, we knew it before we started, but we didn't appreciate how much it is until we did our focus group is how much anonymity mattered to these citizens. So here's something that when we started analyzing the data, we saw very common. Somebody would say, a teacher never shows up and the government official responds and says, okay, can you give us information about the village and the teacher? And then they don't, they never respond. And so when we were speaking to people, when we hold the focus group, it turns out that when they heard back, they thought that the anonymity has been compromised even though that was not the case, okay? So many thought that if the district government can respond to the platform, they know who they are. And so they disengaged. So for us, the lesson was that we need to strengthen anonymity. I think there's also, we learned that there's a bit of a tension between all-purpose platforms and specific platforms. This platform, we didn't limit people in like, you can report education, health, water, whatever. There's some studies that suggest that if you do platforms that are very only on water or only on health, you can get more actionable messages. So I wanna throw that out there. In terms of the district official, they were really happy. They were like, oh, this is great. We totally love it. Yet, it also was clear that like without a major involvement of the NGO that had like an intern that sit in the district government and basically nudging their officials to respond to incoming messages, this would not work. And so I want us to also be mindful of that. It's also clear that there was a mismatch in the between what citizens expect and what government delivered. Citizens were expecting the government to take action. They weren't expecting them to say, oh, the right person to go to is the principal or the PTA. Some additional lessons that I'm gonna end here. The government, the platform was more techy than what the local government used it, you know. The messages could, you know, you could tag them as like pending and complete. And, you know, there was a way of sending messages back to citizens making complain, updating them on the status. Oh, we're investigating, oh, it's pending. Oh, we checked it and this is what we found. They haven't engaged in that. It wasn't used in this way. We think that for these types of platform to work, you have to absolutely have to update citizens at every stage of what's been done with the complaints. This is kind of a bit unique to Uganda that the importance of including or so lower government level. One of the things that we didn't do well and I regret is when we spoke to facilities, they had no idea what messages came in about them so they didn't get any feedback. So teachers, so principals didn't know when parents were complaining about teachers. Facility in charge in the facilities had no idea when complaints were made. Unless the government contacted them and that is not a good idea. Okay, just in conclusion, we think that these types of platform, we don't think that we need to shut down all type of these platform. I think there's some promising things in trying to utilize them. There is clearly a demand and we believe that there is maybe a way of setting it up in a better way than what we've done. The most important thing is that we were working, most people are just with like flip-down, there's like old Nokia. In a world in which you have smartphones, which is a world where we'll be in like five years also in these areas, I think that's gonna be a big difference because you can like geotag yourself, you can take pictures, it's just you don't need to report where you are, you just basically geotag yourself and so that's gonna be a sea of change. And the last thing I would say is that here I'm just reporting the results in terms of service delivery. We think that it's also important to understand take-up because there was a lot of variation in both within and between villages. Some villages used it and some not. In some villages there was really high uptake and in some villages there were not at all and so we have another paper that tries to explore using some network analysis, explore what villages and what villagers are more likely to use the platform and if people are interested, you can send me an email and I'm happy to share this with you. Thank you very much.


Welcome back. In our last lecture we have done a very intricate analysis of computing the number of the square roots of 1 in Z mod nz. It was first of all factoring n into several prime powers and then solving the equation x square equal to 1 modulo these prime powers that gave us linear congruences mod each of these prime powers and then we got a set of systems of simultaneous linear congruences which have unique solution modulo the product of all those moduli which is your n and that is by Chinese remainder theorem and so we can compute the exact number of solutions to x square congruent to 1 modulo n. I would suggest that you takes n equal to some number like 30 or say 48 and actually work out all those things quickly and that will tell you how that intricate analysis works in some very particular cases. Now we are going to take our attention to somewhat different type of mathematics. In number theory there is a function called the factorial function. This is one symbol that is lifted from literature which is the symbol of the exclamation mark and we use it to denote the factorial. So, a factorial is 1 into 2 into 3 into dot dot dot all the way up to a that is our factorial function it is a expanding function as a goes to infinity a factorial goes to infinity much faster than any polynomial in a. However, when you look at the z by nz or in particular the z by pz the factorial function is not going to increase because when you look at z mod nz you take powers or you take products you are allowed to take the multiples of n out. You after all have only finitely many elements in z mod nz you have exactly n elements in z by nz starting from 1 to n or if you want to start from 0 then you go up to n minus 1. So, it is interesting to ask what would happen when you take the factorial function in such a ring and this is a very interesting result that we now have this is called Wilson's theorem. So, the theorem says that when you take a prime p and you take p minus 1 factorial. So, that means you take the set z mod pz remove 0. So, you have all the non-zero elements there 1, 2, 3 up to p minus 1 if you include p that p is 0. So, we are not going to include that back. So, you have only the non-zero elements 1, 2, 3 up to p minus 1 p minus 1 factorial is the product of all these numbers and it is interesting to note that this is equal to minus 1 when you are working in the set z mod pz. Now, this very important and very basic theorem is given the name of this mathematician John Wilson. But as it happens often in mathematics or in several branches of science this is not originally due to Wilson. In fact, it is not at all due to Wilson. This was stated in 10th century AD. John Wilson is a mathematician from 18th century. This was known in 10th century AD and in fact, Wilson did not solve this at all. Wilson's teacher Edward Waring he gave this as a problem. He computed this p minus 1 factorial for several primes and he noticed that this is congruent to minus 1 mod p. So, as it is very common with teachers Edward Waring gave it as a problem to John Wilson to show that p minus 1 factorial is congruent to minus 1 mod p for every prime. But Wilson could not do it. However, it was Lagrange who did it in the next year 1771. But since it was Wilson who made it well known who posed this as a problem when he received it from Edward Waring and because he could not solve it he asked people and so it came to be known as Wilson's theorem. The proof of this is very easy and it will draw upon the thing that we have done earlier that the square roots of 1 in z mod pz when p is prime are only 1 and minus 1. Remember those calculations or remember that result that is a result that we are going to use here. So, we go towards proving this statement that p minus 1 factorial is congruent to minus 1 modulo p. So, note first a equal to 1 and minus 1 are the only solutions a square or we should perhaps say x square congruent to 1 mod p. So, what it means to say is the following. Thus, if you have a from this zp or zp and you take a to be different from 1 and minus 1 then the solution to ax equal to 1 in zp has or the solution to ax equal to 1 in zp is not equal to a. Let us think about this statement again. We start with a non-zero element in zp. So, we are looking at elements 1, 2, 3, 4 all the way up to p minus 1. We fix 1a from this set and we pose this question what is the solution to ax equal to 1. Now, this is same thing as solving ax congruent to 1 mod p and since you have taken p from 1 and p minus 1 the GCD of a and p is 1. Therefore, you have one solution and you actually have a unique solution. The only question we are interested in is whether this solution can be equal to a itself. We know that ax equal to 1 has a solution. So, we know that there is an element b again in zp such that a into b gives you 1. This is something that we have done quite often you know when we looked at a equal to 5 let us say and p equal to 7 and we wanted to solve for 5x congruent to 1 modulo 7. This had occurred in one of the solutions of linear equations, linear congruences or while solving the simultaneous systems of these. And we observed that 5 into 3 is 15 and 15 is congruent to 1 mod 7. So, the equation 5x equal to 1 in z7 has solution 3 which is different from 5. However, if you are looking for 6x equal to 1 modulo 7, 6 is itself the solution and the reason for that is 6 is equal to minus 1. If you were looking for solutions to x equal to 1 modulo p, x equal to 1 is the solution. So, when your coefficient a happens to be 1 or minus 1 that a is the only solution. But the previous analysis of looking for solutions of x square equal to 1 modulo p when p is odd prime told us that there are no other solutions. So, ax equal to 1 whenever a is different from 1 and minus 1 whenever there is a solution you will have that the solution is different from a. This is one thing that we keep in mind. There is another thing that we should remember. Further if b satisfies ax equal to 1 in gp then a satisfies bx equal to 1 in gp. So, you have earlier we have seen that ax equal to 1 has a solution other than a call that solution to be b. Then you can ask the same question for b. So, you have bx equal to 1. This is another solution that you have created that has solution to be the earlier a. So, going back to our example when we had 5x equal to 1 in g7 we noted that 3 was the solution. But then you ask if you ask for 3x equal to 1 in g7 5 will be the solution. So, all the numbers from 1 to p minus 1 other than 1 and minus 1 can be put in pairs where a will be put up with b where the property of a and b is that a into b is equal to 1 modulo gp. This is what we are going to do. Thus all of the elements I will not take 2. So, I will not take 1. So, I will start with 2, 3, 4, 5 up to p minus 2 can be paired a, b such that a, b equal to 1 in gp or a, b congruent to 1 modulo p. So, when I take the product p minus 1 factorial this is 1 into 2 into 3 and so on up to p minus 2 into p minus 1. So, these elements have been paired as a1 into b1 where a1 and b1 have the property that their product is 1 in gp a2 b2. So, on up to a, r, b, r. Remember this is an even set even number of elements. This is because your p is odd. So, 1 up to p minus 1 are even elements and you have removed 2 namely 1 and p minus 1 from them. So, what you get is an even number of elements and therefore they can be easily paired and finally we have this last element p minus 1. Note that all these products are 1 and finally we are left with p minus 1. Now we are almost there thus p minus 1 factorial equal to 1 into p minus 1 which is really minus 1 in the set gp or what we have is p minus 1 factorial is congruent to minus 1 mod p when p is an odd prime. So, the proof is quite simple but remember that the intricate analysis that we have done for the number of x square equal to 1 modulo n is one of the major steps behind the simplicity of this proof. So, we have such a nice result and you may ask what would happen in general. So, clearly Wilson's theorem does not hold in general. In fact, what that means is that if you take a number n which is not a prime, you may ask you know earlier equation slide said that p is odd if you put p equal to 2 then yes that also holds p equal to 2 here p minus 1 is minus 1 but that is also 1 modulo 2. So, there is actually no statement for p equal to 2 but we may say that p minus 1 factorial congruent to minus 1 holds for all primes that is something that we can say and when we ask for the same question whether it holds in general we are asking for a number which is not a prime. You may look for n equal to 4 and you will see that n minus 1 factorial which in the case of n equal to 4 is 3 factorial. 3 factorial is even number so clearly it cannot be minus 1 modulo 4. So, you already have that whenever n is not a prime that n is a composite number and then the smallest composite number 4 gives you that the equation does not hold. In fact, you can say more we will say so if n is not prime if n is not a prime then it has a factor which is not equal to 1 and not equal to n by the very definition of primes but you can also get that factor to be itself a prime. So, when you have this factor and if you have that n divides n minus 1 factorial plus 1 which is to say that this equation holds instead of not holding then q divides this because q divides n but q is less than n so q is less than or equal to n minus 1 so you have that q also divides n minus 1 factorial this will be 1 into 2 into dot dot dot somewhere q will have to appear and then you go up to n minus 1 and this is the contradiction. You have that q divides n minus 1 factorial and also that q divides n minus 1 factorial plus 1 that cannot happen. So, on one hand you have this on the other hand you have this which is not possible. So, clearly whenever n is not a prime we do not have Wilson's theorem but in fact something more can be said in fact if you have that n is not a prime then almost always we will have that n minus 1 factorial is 0 mod n n will divide n minus 1 factorial. This is something that always holds except 1 or 2 exceptions. So, I will what I will now give this to you as an exercise I will not give you the solution neither will I give you any hint to solve this problem. Take this as an exercise and find the exceptions. By exceptions I mean the situations where you do not have it to be 0 mod n. So, almost always is not the same as always almost always means except for a few cases you have this. So, those few cases where n is a composite number but n minus 1 factorial is not divisible by n those are the composite those are the cases that I want you to find and in the remaining cases I want you to prove that n minus 1 factorial is congruent to 0 mod n. So, this tells us that there is something very special which happens when n is a prime number the arithmetic in G p has some special properties compared to the arithmetic in a general G n. So, what does it mean let us go through these things once again if you are looking at linear congruences modulo a prime number then we see that the congruence A x congruent to b mod p has a solution when a comma p is 1. Now, p has the property that when you look at G p only the 0 is the one which does not have G cd equal to 1 with p but all other numbers have G cd equal to 1 with p and therefore, A x equal to b mod p will have solution whenever A is not 0 in G p if you have p dividing A but p and of course when because the G cd is 1 we have a unique solution further if you have p dividing A but p does not divide b then there are no solutions. So, you will have 0 x equal to a non 0 b congruent to p cannot give you a solution if you have that p divides A and p divides b then everything is a solution because this is like asking for 0 x equal to 0 mod p and whenever you put anything for x you are going to get the answer to be 0. So, barring the cases where p divides A the interesting cases are where p does not divide A or when you are looking at the elements in G p you are looking at the case where A is not equal to 0 whenever A is not equal to 0 the congruence A x equal to b has a unique solution. This is something which happens for complex numbers this is something which happens for real numbers this is something which happens for rational numbers also. So, whenever you have any complex number A and a complex number b and you ask for A x equal to b if the complex number A is not 0 you can simply invert that and you have the solution x equal to b upon A or x equal to b into A inverse or A inverse into b the multiplication is commutative. Then we would say that alright for G p there are things which are happening like the complex numbers or real numbers but for complex numbers and real numbers you have one more interesting thing that is that when you have a degree D polynomial any such polynomial will have at most D roots this is something that we have again used this is called the fundamental theorem of algebra. It says that if you are looking at a polynomial in degree D in one variable over complex numbers it has exactly D roots if you are looking at a polynomial over R you may have a smaller number of roots because some of these roots will actually be in C they will not show up when you look at only R. So, for R or for Q the rational numbers the correct statement of the theorem says that a polynomial in degree D has at most D roots you may have a smaller number of roots or you will have it to be equal to D but you will never have it to be more than D. The condition is that the polynomial should be a degree D polynomial does the similar theorem hold for G p that is the question and yes it does hold for G p and the theorem is theorem due to Lagrange you will see that Lagrange has done quite a bit of fundamental work in number 3 among other areas of mathematics and this is also a theorem due to him. So, he says that if you start with a polynomial of degree D over G p over G p means that you have coefficients coming from G p that is what it means then any such polynomial has at most D solutions in G p. Before we go to the proof we wonder what would happen when you have other cases or is there anything that we need to be careful about. So, number 1 note that the number of roots can be smaller than D. So, there are indeed examples where the number of roots can be smaller than D. For instance if you are looking at x raised to 4 congruent to 1 or to say it in other words if you are looking at the polynomial x to the 4 minus 1 in let us say G 2 this has only one root namely x equal to 1. But here you may say that G 2 actually has only 2 elements 0 and 1 0 to the 4 is 0 and 1 to the 4 is 1. So, that the fact that there are smaller than 4 roots you have only one root which is smaller than 4 is not very striking because G 2 alone has elements which are smaller than 4. So, if you are looking at a polynomial of degree D which is the degree being bigger than P then it should not be surprising that you have the number of roots to be smaller than P. So, if I give you an example I should give you an example where the prime P is bigger than the degree and yet we have smaller number of roots and so we will look at x square plus 1 in let us say G 7 has no root. So, x square plus 1 in G 7 has no roots you can actually quite quickly check this you have 0, 1, 2, 3, 4, 5 and 6 and let us just compute their squares. So, 0 square is 0, 1 square is 1, 2 square is 4, 3 square is 9 which is 2 and 4 is minus 3 so its square is 2, 5 is minus 2 so its square is 4 and 6 is minus 1 so its square is 1. So, there is no solution to x square plus 1, x square plus 1 is same as x square minus 6 when you are looking at G 7. So, there are no solutions at all your degree of the equation is 2 your number of elements is G 7 is 7 you could have had 2 solutions 1 solution but this solution this equation has no solution. So, this is indeed an instance where the number of roots is smaller than the degree D. What if we work with something which is not a prime does this result still count does this result still hold? No, it is important that you work modulo of prime. If you take n equal to 8 and we have solved the equation x square congruent to 1 mod 8. So, x square minus 1 in G 8 has 4 solutions 1, 3, 5 and 7 you have 4 solutions which is bigger than the degree. So, indeed it is very much important that you work modulo of prime if you work modulo a composite number then Lagrange's theorem need not hold. We will see a very simple proof of Lagrange's theorem in the next lecture and then we will do some more analysis with the set G P. So, I hope to see you again in the next lecture. Thank you very much.