The following is a presentation of T-F-N-N. The Tiger Technician Hour. With your host, Basil Chapman. Call now, toll free at 1-877-927-6648 internationally at<PHONE_NUMBER>. Now, Basil Chapman. Everyone, Basil Chapman. Yeah, this is the Tiger Technician Hour<PHONE_NUMBER>, July 1. Can you believe we started the second half of the episode? Just goes like that. Nothing. Boom. So, as we're looking at this, I had a question about the queues. That's the NDX100. Could there be a different count? You know, I don't really get a different count. I do this innumerable times. And if the queues trading at 189.30 right now, up 2.56 in daily leg D, under the previous high of 191.32, I don't like that. I like to Ds to be above the previous highs. And the same thing in the weekly chart, you've got a peak C. So 191.33 stars legs D in the weekly chart. You can call it D slash B because that is a peak. But really, I think D is going to be what we call it. And no, you can't call it D. It has to be D. And in the monthly chart, I just don't see anything else. So if you can come up with something like Mino or send it to me or discuss it, call in whatever it is. But this V-shaped pattern closing above 187.53, the high of October. That was back in, actually it was two months. We went above it. But only one month we closed above it. That was in April. Pulled back for 84.74. Let me go back. Let me go back. Let me go back. Let me go back. Let me go back. Okay. So let's look at this. Yeah, we'll do it together. Look, you got a peak D at 140.79 back in July of 2015. 2015, it's four years ago. And then there's a sharp pullback to 84.74. Then it goes to a higher high. So I call that E. It could be an A, but it makes no difference. Whatever it is, it had a chamber of Roman Canon and a sharp pullback after that. But it only pulled back to 94.84. So it went to a slightly higher high. I believe it went to right here. 140.79 was the high of July. And I think it was 115. I can't see. 115.75 was the high in December. And then a pullback very sharp. 2015 goes to the low of 2016. 94.84. But either way, you see this A right here which becomes a gray A at first because it's under that previous high. This becomes a restart. That's the whole thing in my chapter in the CD introducing the Chapman Wave methodology. But then when it breaks out above that 115 high of December, that starts a leg B. No matter how you count all this, even if that was an A, this would still be a B. So it's an E slash A if you want to call it. Then it goes to an A minus because this is another A over here lower down. You remember, you've got to count each peak and each trough. That's how you do it. That's how you do the navigation in the Chapman Wave methodology. So this leg B right here in August was it? So 119.22 and 119.66. So the October 2016 high is B. Then it goes all the way up, all the way up to a C. That was in 2017 and that was in, 2018 and January. Pulls back for a month. And it goes, remember that horrible month that first week, first two, three weeks. And then February has a nice rally. And then it goes March. Goes to a high of 175.21 and that's D. Then it goes to E and F. Everything works perfectly. There's just nothing wrong here. We managed to get that top exactly in the Dow and pull back very sharply. Goes to 143.46 in the queues. Went from 187.53 in October of 2018. Down to 143.46 in December. And now we've got a brand new A or G. I don't know. It's hardly like a G. But it could be exactly the same patterns. Really repeat very often. So this high that was made right here back in December of 2015 at 115.75. That could be turning into a G. And then there'll be a pullback. Maybe there's a sharp pullback. But it's way above the 143.46 high. And then we start a peak A and a peak B. So either way so far this looks like a really good set of chops. I like it. All right. I hope I answered the question. Okay. Let me know if there's still a question involved. Now let me go through this again. This is the beginning of the month. We've got the end of the month. So we can talk monthly charts here. We've got 26,722. Only up 122 after being up much higher. It was at 26,890. 170 points higher earlier on. Still didn't go to LAC. And that's the reason why I'm saying that this Chapman Wave inside track repellent zone such an important in my work. I have just certain techniques that we repeat over and over and over. And each one of them has been developed not over years and sometimes decades of practice, of due diligence, of correcting errors that I've made when everything said that there should be a pullback. Let's call this an inside track repellent zone. It says there should be a pullback here. It says if you break and close above this dashed green line, you've now broken out. In this case it was to be a laying seed but it was to be above that trend line. It means now this whole area of resistance becomes support. These are techniques that I use all the time. MACD, look at the MACD, it's still very strong. It's forming an M shape formation. It's beginning the process of running out of steam but it hasn't run out of steam yet. And at the same time Stochastic says, whoa, under 80% and 67%, that's not too good. So there's a divergence. So another reason why I say the upside I think is limited we will squeak towards new recovery highs in some areas. It'll be all-time highs, maybe even in the Dow. And then I think we've got to be prepared for some kind of a consolidation. All right, cut that out the way. Done a lot of work. Let's just run this real quickly. Give you parameters. The Dow needs to hold 25,530 support this week. Anytime it closes below that, that's a real problem. S&P is trading at 29.63. Up 21 has gone to a leg C, has gone to a new all-time high, has gone to a leg E in the weekly chart. Ooh, interesting, isn't this, huh? And F slash A in the monthly chart, everything about it points to it probably being a new leg A. Very powerful breakout from 23.46 to today's higher. 23.46 to 29.77, unbelievable 23.46. And this is still a leg A to the upside and we're halfway into the year. We're actually starting the second half of the year, July. And there hasn't been a peak yet. That is very positive. That is very positive. All right, let's get on with the story here. So S&P, all I'm going to say is that the 29.48 to 29.42 is probably an area that needs to hold this week. If there's a pullback below that, that is really not good action. I just did the cues, but I'll do it again. QQQQ trading right now, up to 2.70. Almost at the low of the day, but it did gap up huge. I think that gap will be full, but it doesn't have to be full. The need to do it can sweep to slightly higher highs in the interim. I'll be right back. Puzzle chapter tells us 127. S&P's up 22. Love to take your calls. I'll be right back. 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I've seen this before. I can't even remember. I think we once had it on my list, but we never did anything about it. Or maybe we did. I just don't remember. It was a long time ago. I like it very much. But here's the issue. It's in leg D, second leg D in the weekly chart, leg B, only leg B in the monthly chart. I haven't had a chance to do the count in the daily, but I love the look.  We're having the same spike to the upside. The only difference is that in this case, the stochastic is very strong. The OBV is at high. So I love this at 43.55. Since you're a very long-term position player, I would normally say, look, yes, it can go higher. It could even go a point, a point and a half higher. But if there's a pullback at this stage for whatever reason, it could get stuck for another two, three weeks and then pull back to the 42.5 area, a point down. It's a big deal. If it's a point down and a point up, why not take a nibble? Well, the high, I think it's the all-time high was at 46 something. Let me just check with the prices, maybe less. Yeah, 46.88. Back in December of 2017, it's making a cut formation. My target would be the 46s. So I'm going to say to Piki and the Dan, yes, in your case, I'm going to say to you, start a small position right now, but that's not your position. This is just to get a feel for it because you don't want it off your radar and you're a longer-term player, and I think it's going at least two points high, which is 5%, which is fine. I don't think it's got more than 3%, 4% or even 5% to the downside. I think it's more time. Look at the way it makes these stair-step moves and then it pulls back. So, yeah, nibble. Just nibble here at 43.56. This is not your real position. This is just because you're interested and you really would like it and it looks great. The monthly chart is good. The weekly chart, it's above the 9-period moving average. This is BIP trading at 43.56, and we call it Brookfields, Infrastructure Partners LP. I do like it. Last time it had a D, it just pulled back modestly to the 9-period moving average. That's at 42.45. So, yeah, you've got yourself a 1.2, 1.25-point risk, but start here and then let's look at it again together if it can pull back to the 43. Sorry, 42.50. I'd love to see it at 42.20 with a chance of making one more slip to the downside because then I think it fits into the pattern of sell the highs and buy the lows in a channel formation. That's on a short-term basis, longer term, it says buy every pullback, but that's not my belief right now. I think you need to just let it play out because this is the second move in the weekly to a D, so now I get a little more cautious. Okay, so that's that one. And Tul Ray, is that what it's called? Tul Ray, Tul Ray, Tul Ray? Yeah, Tul Ray Inc. This is in the, I don't know if it's in the medical cannabis area, medical cannabis. A wonderful example of the Chapman Wave Eiffel Tower, straight up, straight down, capital A pattern hit 300 round number in September of 2018. Had a little bit of a problem, went down to the 34s, trading right now at 47.69. So the question, where was the question? Where was the question? A gold 25 in the debt. Yes, so I don't know if you've played this before, and I say play because it's really different because it's really difficult to position yourself in such a long-term decline. A couple of years, well, a year and a half. I don't know, it's only a year. But look at the move to the downside. You're long. Okay, that's what I wanted to hear. If you're long, I suspect, and you probably know this, Goldie, you can see that there's a really good chance that if it can take out at 47.64 right now, if it can close two sessions, I want two sessions. I want you to know it's not just a one-time blip, and it needs to close above the high of 48.48. That was the high of the 27th of June. Then I think you're looking at a leg D in the next two weeks above 51.03. With that said, the MACD is strong, but the stochastic pullback quite sharp, and that weekly chart is just repairing a lot of damage. So it needs to hold a 45.50 area. That's a two-point. I don't know what kind of stop you have, but at any point if it closes under 45, yeah, let's call it 45.50. If it closes under that, I would just say hold off, not just now, but you're probably going to have to hold off for a few weeks. But at this particular moment, I like the action. Let me just look at the 120-minute chart. Yeah, the 120-minute chart confirms what I said. It needs to get about 48.48, and then it's going to look a lot better. So that's all right. Now, I've got a couple of things that I want you to do. Just real quickly, the IWM, has gone to a leg E with a red candle, even though it's making it higher, higher than just the other day. It's up 77 cents and 156.27. This is going to be a problem. Is there such a kind of, is there rotation going on? I almost had this last week to say, you know what, there could be some crazy kind of rotation here, and you could finally see a leg D in the weekly chart of the IWM, the Russell 2000, and then the more I looked at some of the stocks within it, it's just a real mixed thing.  So it's in the good category here, but it is red. It's a red candle, having come way off the high. So I'm just going to say, the IWM is acting much better over the last three sessions. Not good enough, but much better. And the reason why I mentioned it is that Thursday, I thought, maybe Friday, just to have it long, because it was so oversold that it needed to play catch up. Now it might have played catch up. So we're going to be watching this closely. But I just say, Russell 2000, at this particular point, there's a lot that needs to be done to improve the technicals in the certainty in the daily chart, and the monthly chart needs a lot of work. So that's that. Okay, here we go. Crude oil. Crude oil right now is up 48 cents, 58.7. Let me read this if I can find it. Maybe I'll have to wait. So last week, at some point, Bob sent me, he sent it to a couple of people an email and says, just heard a news brief, a news brief on a meeting by the EU members to attempt to have Iran not exceed the enrichment quotas set in the nuclear development agreement. It may be that EU buying oil from Iran may be an attempt to settle the agreement on nuclear development. So the big question he has is, came later on, he sent me a questionnaire question. No, wrong Bob. Let me find it. So the question was, is crude oil going to really come down sharply? Is there a glut? Is that going to be shown up as in the price point because it keeps getting weaker and weaker? No, I'm looking at crude oil, and I think crude oil is in an area right now that says, and it's actually very interesting because I'm stalling because I like to hold off until after the break before I actually tell you my conclusion. And I'll tell you back in a moment, the Dow is only up 95 now. It's a piece of 18. And we'll talk about crude oil and the whole put-purry of different sectors. And I think crude is going to be important. So I'll be back in the talk about crude. You'll have to come back to hear what I say. Since 1984, Basel Chapman has been using the Chapman Wave methodology to advise traders of his expert market opinion. While originally hand-drawing charts from the late 1970s into the 1980s, Basel noticed that prices under most circumstances virtually always had a certain number of legs to the upside before declining sharply. 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Let me just go back to where I was if I can find that. Yeah, crude oil. So crude oil is held very nicely. It's stuck against its 200-period exponential moving average of $59.82. $59, yes. It's trading at $58.98, of $0.51. In legs, C. Magdian's stochastic are good enough to say it could go higher. I said this about a week ago. I believe that crude oil should trade between $61.75-ish. This is the continuous contract as resistance. And $57.70 as support. We're in the middle of that right now. I'm just sort of the lower half of that right now. And that's kind of what I'm looking at. I don't think it's going to break down not yet, but I do see that the upside just at the moment is limited. If there is a close above $62.50, that's something completely different because now, all of a sudden, the 60th use to the 64s becomes a magnet. I think that's going to be a bit of a problem just on the shorter term. Let's go to the gold now. The gold itself is down 19 at $13.94. It made a peak effort, $142.90. Remember, there's a continuous contract. So I must just double-check to see if that wasn't smoothed out and changed. $142.90, that was just six sessions ago, five sessions ago. And now what we're looking at is tested the 14-period moving average to make these goods. The castings pull back under 80% and 73%. The technicals in the weekly chart after the Chapman Wave Cup and Ladle breakout are still very favorable. They still look very good. And it's only a leg C in the monthly. I suspect that in this particular period, let's just look at silver. I'm going to go through a couple of things at the moment. Silver right now is just down a little bit. It's down 0.6%. It's not a little bit. $15.24, down 0.09. Had that peak C. Well, it almost looks like Tul Ray there. It looks as if it wants to go now back. Remember, I said, watch this line. I said this week ago. I said, watch this 200-period exponential moving average at $15.35. Why? Because I think you're going to go over and under and over and over and over and turn it into some kind of a magnet unless you break decisively away from it and it was close today. But if it goes to $15.10 or less, that's a breakdown. If it can hold here nicely and have a little bit of a pop to the $15.35, 200-period moving average again, it means that's a magnet that's going to hang around. Even if it goes high, it's going to come back. Even if it goes low, it's going to come back. I want to see this as a magnet to see, I want to talk about it as if it's a magnet and to see if it remains in this rectangle formation. Limited upside, limited downside. If we go to EURUSD, the Euro has made a peak D with a very sharp pullback. There's that rectangle. Finally, it broke that rectangle to the downside that I said before. The weekly chart looks okay. I didn't like the doji candle close on Friday. So yes, it closed on a daily basis that says it's a problem. All right, so that's the Euro. If you look at USDJPY, which is the yen, dollar yen currency pair, a nice leg B to the upside, fourth day of rallying. And I like this just on a short-term basis. At 108.44, it's going to have a lot of resistance in the low 109s. Next thing, we're talking about trucks in the den. The Ford F is for Ford, a leg E to the upside today, pulling back after that intraday spike. Now it's full of the gap and it's at the low of the day. I've been impressed over the weekend looking at Ford General Motors. Look at this. I mean, they're not breaking down. They're holding well. So there's this rotation that's going on throughout the market to say that some stocks, for instance, look at Intel in the semiconductors. Intel trading up 9 cents today, 47.95. But look at applied materials. Applied materials spikes up at the low of the day right now, and it's still up 72 cents at 45.63. And it's screamed to a leg E in the weekly chart, leg B in the monthly chart, and leg C, just a leg C in the daily. So it's saying that even in the semiconductor area, the SMH is having spike to a leg E to the chaff rate inside track. So this is the inside wedge resistance line, this green line. He didn't exactly pull back. Magnet is good. Stochastic is good at 84%. So I'm looking at this, and about three weekends ago, two weekends ago, I showed the strength of the monthly chart to say that even at the low, when it plummeted from 114 down to 80, in December, the price didn't drag down the nine period exponential moving average. And it's rallied, and that's really a big positive, and it remains a positive, even if there is a pullback from this gap up high, to say, hey, we're not quite ready to do the big stuff right now. That's a fantastic move, going from 97 to 115. We have not been long. We covered our shorts. We were short from the 100, off the 120.71 high in April. We got shorted 116. It went all the way down to the 98th, 97.61 was the low. Down to 98, we were already covering almost the last position, then the last position was at about 305. And then I should have said, you know what, there must be an oversold balance. I said it, but I didn't think it would last. This has lasted a lot longer than anticipated, but I have to tell you that that monthly chart is really impressive. And it says that later in the year, you might see the SMHs testing and breaking the 120.71 all-time high. Question about EEM, EEM right now coming up once, coming up twice. 43.34, leg E to the upside. Oh, it actually looks like the semis, doesn't it? D, E. They'll kind of go together, I guess, international stuff. Yeah. So this is, yeah, this confirms for me that I think there's some limited upside for the markets generally. There should be a little more. The wave, in other words, it's saying high tide is coming in sometime in the next day or two, but the waves don't know that, so those waves can go a little bit higher before we start some more of a consolidation. I think July is going to see quite a choppy month, and this time we might see a couple of sharp down days rather than these big up days as we do some consolidating, but I'm still a mega ball in terms of outlook for 2019 going into 2018. But in the meantime, back at the ranch, let me go through something else. Question about the financials, yeah, it was just, okay, financials. The XLF is trading at up 27 at 2788. I like this. I spoke about it on Friday. It was breaking the Chapman Wave inside track repellent zone. It spiked above at 2814 was the high of the first of May. Now it's the first of June the highest 2805, 9 cents off that high, and I do think it's going to go to a leg D and that'll make a leg D at 2815 in the weekly chart as well. Very subtly, we start up with, oh, finally, let's just look at that. We've broken out for the very first day of July. We actually broke the down channel inside track repellent zone. We're above it, except it's three hours into the first trading day of the first trading day, first trading week of the first trading month of the second half of the year. So, all I can say is this is a nice start, but that's nothing. These are hours, not a month, not a month's worth of trading, but it's a really good start to say, hey, finally, there's a push against their trend line and it hopped over it. Let's see how the month concludes, but yes, that's why we in a bank stock done very well. We're in other areas that are doing very nicely that we'll see how this one's going to turn out. But in the meantime, I wanted to also answer a question here about oh, let me just show you this here. This is the 10-minute e-mini going towards the 200 period moving average of 2960 to 2964. I'll be back. 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Distributor Four Side Fund Services, LLC. Don't forget, you can listen to TFNN 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 for the latest market information. So I thought this was very interesting. For my subscribers, most of my subscribers know that I've been talking about the skyscraper phenomenon for decades, not just recently, but really for a long time in the 1980s. So in Boston, one Dalton, Boston's commanding new skyscraper, Conjures Architectural Magic. That's building right here in the middle. Boston, this is the Prudential Center, the Christian Science Center is over here. This is a property right opposite the Sheraton Hotel, Park Plaza Hotel. So this is gone up. All of a sudden Boston, which was the college town of the Northeast, becomes really a pretty, it's a major city. And if you know the roads, the time consumed going to work is every one of the records in the country. It just takes a long time. So I showed my subscribers this and I've been showing them buildings for quite a while in the Boston area. But this one here is very interesting. I actually like this design. I've always hated this Prudential since I got you. I called it one of the besides three other buildings. This is the Prudential, it was the library, the new library. Now they're finally off the decades. They've done something to the front of the new part of the library to help and ameliorate the ugliness that it was with all that concrete. And then the building city hall was just one of the worst. And then the Heinz center. I love the, can you see it? No, it's not even here in the picture. I love the John Hancock building. I've got pictures of it as the world's tallest plywood building. And then they had to put the glass back because they had to replace all the glass. Even after it was finished, they did all the glass over. It hadn't been opened yet. So this is I love this design. I think it's terrific. That's not the point. The point is above the turnpike. Now what they see, this is what cities do. They've never had so much income coming in. Taxes and incomes are improving. Everything's improving. Housing, boom, real estate, yes, everywhere you look. But the spending boom is unbelievable. And the government, the federal, whatever it is, the cities, each town, they're overspending. Well, there's a great article this morning I was reading. It just goes through basically the spending. That money's there. It's just being spent excessively. I mean, these things are important, but it's excessive. So my question to my subscribers, to myself really is every time, as long as I've been in Boston, which is a long time, at a certain point when they start talking about hotels that are new hotels needed for the world, we'll keep moving it from the World Trade Center wherever it's going to be. I say, oh my God, we've got another couple of months and then we go into another recession. This has gone on a long time. And the way I'm looking at it, especially if you're looking at New York, Brooklyn, New York, and you go all the way through Northeast, when I go through Connecticut, I see the same thing on the way to New York. I have to tell you, I don't know when this top is coming in real estate, but when you look back over a period of time, and I've done this ever since I've been, whatever city I go to, when I suddenly see a very tall building or a massive construction, I always want to know what was the time period? Why did the council allow that to go through? Because it's always at a special time that they say, oh, you know the income, 120 stories and oh yeah, we've got to let that go through, but there's a casino here in Boston, right? Like the Angkor Casino. It all gets passed, but there's a lot of money that's involved. And at a certain point, you look back and you say, oh, wow, that was like the top of the market. They allowed that because they wanted the income, but they were being greedy. It's hubris. And I'm looking at this, I'm saying, where's the hubris? Where are we? I think we're getting closer. I don't think it's just yet. I think there's still some time, but we are absolutely getting closer and closer to some kind on the northeast here of a pretty decent construction decline. And I don't when it's coming, but I think we're getting closer and closer. I wanted to show this because it's part of the history, part of what I do for my subscribers to my opening call besides all the other charts over the weekend that I show. I mean, just everything from analysis. Let me see if I can find one right here. Right, right, right. Yeah, that's my Dow analysis. The Dow daily is in a buy mode. The Dow weekly is in a buy mode. The Dow monthly is in a buy mode. And I give you all the parameters for daily weekly monthly. And in sum, this is what we're looking at. So it's a very detailed report. And as I say, we've had some really good successes managed to pick just the day before the high of the Dow in April to go short. And we managed to pick the exact day to go long during the third. And we've had some really nice trades, a nice position play. So if you're interested, my opening call that's my the opening call. That's what it's called.  You'll see you'll be able to get it for free for a month as a trial.  So now we've got the XLF acting very nicely. The IAA, I've spoken about this before. This is the what is it called? It's called the iShares US broker dealer and securities ETF. This is very important. What I've been saying for a long time is that if it starts to trading the 65, 64.57 was the all time high most recently. And but if it starts to trading the 65s to turn the 63s into support and they can push even a little higher. That's going to tell me that people are buying the brokerage houses are starting to see money come in and that people are starting to trade because now they're getting the volume. And if you look at the CME the CME is in fact the Chicago Board this is the Chicago Mercantile Board. This is the exchange and it made a higher in the 10th of June at 204.56. It just went straight up. Look at this weekly chart. It goes from the 162 area what was that? 161.05 in March of 2010. It makes one little minor peak and goes to a peak B all the way to 204.56 and that's only peak B. There's no other count that should go to a C and a D. So the Chicago Board is taking a bit of a breather here and that's telling me that more and more people are becoming interested in the stock market. I think that's really important. And that's very important for the mega bull to really pick up steam later this year. Okay, cut that out the way. TLT bonds. Now here's the query down a dollar three at 131.78. Made a peak F in the Chapman wave and you see this 132.58 high of June the third. Let me just put that in as June the third here. So that is 6 3 and you see the high that was just made 7 days ago fractionally high 133.51 I put it in red because I thought this is pretty serious on the 20th of June. Now we're going to see something very important. This particular week is what we're going to see if this works or not. Look you see this see the way the Magdeus stochastic was so strong in the June high but look at this we come into if I can find it right there we come into the June 20th high 133.51 a higher high in leg F and look how weak the technicals are price hasn't fallen much but I'll explain something when we get back Puzzle Chapman Tiger technicians are talking about this I'm certain you are or strive to be one of the best of the best at everything you do in life it's the most common trait that we tigers and tigers share. If you're looking to become the best of the best when it comes to managing your money let me teach you to do what most wealth managers tell you can't be done which is how to time the markets. I'm Steve Rhodes author of Mastering Probability and for the last 12 months Timer Digest has been tracking my newsletter signals and me the ranking as their number one market timer in the nation for the S&P 500 for the last 12 6 and 3 months. Timer Digest also ranks me as the number one market timer for gold as well. The fact is markets can be timed and I'll teach you the exact set of tools that I use that has transformed me into one of the best at what I do. Sign up for Mastering Probability today by clicking on the newsletter tab on the homepage of TFNN.com and get immediate access to workshops where I take you step how to use an extraordinary set of tools as well as provide great market calls to sign up today. 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Stay tuned for another great hour of the Trader's Edge heard here at TFNN.com I've always been back so 1.03 down in the TLT and the Lehman 20th Treasury Bond Fund the MACD and the CACACO way weaker than it was back in the beginning of June on balance volume is still pretty good but pulling back so this has to mean that we could be stuck in this rectangle formation but a close at the end of the week holiday week but we are here Friday. Here's a market Friday if on Friday at the close the TLT is trading $1.30 and this is what I said to subscribe when I showed my weekly triple yield chart I said there could be a little bit of a rally in the yields this week I don't know if it should affect the market per se but I can see that. Got that out the way. Question I had was I don't want to know should I add to my spy position. Note to I wouldn't do that but perhaps tomorrow in my newsletter I'll be giving some information about what I'm looking at but perhaps tomorrow we could start the move that goes towards the end of the week to try to make that leg C in the Dow but you're talking about the spy and the spy would be in leg C right now I don't like this red candle but it's still way above the actual Friday close so that is a good sign but it isn't C and then I'm going to say to you I'll talk about it tomorrow but in the meantime I'm just saying I don't know if I would add to it right now I'll tell you a price if it goes to 293.60 you could just nibble on it not to date it has to be tomorrow if you miss it by tomorrow it's tough but I would wait 293.60 and that would be preferably tomorrow and then it has to be an immediate reversal back to the upside now what I wanted to say is that this is so far a high level high in the bonds so it hasn't broken down even though that is a peak F the way I'm counting it right now because the magninians are deteriorating at the moment down below the 14p removing average but I still think rectangle formation think of that if that's a high level consolidation something's going on that's different there are so many things that are going on that are different I'll talk about that tomorrow in the meantime the IYT was a question I had and the IYT is bouncing to a leg E it's still not great action I'd like to see it keep moving higher that's what's moving higher with the Dow going higher I'd like to see that kind of continuity stay tuned for Steve, stay tuned for Dave stay tuned for Tom O'Brien I'll be back tomorrow and please check out my opening call my daily newsletter a one month trial free I'll be back tomorrow


This is an introductory mathematics talk on diaphanetine equations Originally given to the Mathematics undergraduate student association at Berkeley So let's start by just giving some examples So the first one that everyone has come across is Pythagoras's Equation x squared plus y squared equals z squared And you all know the solutions three squared plus four squared equals five squared and There's 18 eight squared plus 15 squared equals 17 squared and so on and The point of diaphanetine equations is you've given it an equation and have to solve it in Integers if we're trying to solve this in real numbers, it wouldn't be terribly interesting We just take z to be the square root of x plus y And So Particularly notorious example is Fermat's last theorem So Fermat changed the exponent here from two to some high in number and asked whether x the n plus y to the n Equals z to the n as any solutions apart from the trivial ones where x or y or z is zero so here n greater than three and He said he had proved there are no further solutions although he probably hadn't and For a couple of centuries, this was the most notorious open problem in mathematics It's just Fermat's last theorem until it was finally solved by Andrew Wiles in the mid 90s So a special case of this is x cubed plus y cubed equals z cubed and Oil originally showed it at no solutions and What I'm going to do in today's lecture is not show that equations of no solutions, but show how to find solutions So this isn't a terribly interesting equation because it's got no solutions to find So instead we could look at some variation of it like x cubed plus y cubed equals 9z cubed and This certainly has a solution 1 cubed plus 2 cubed Equals 9 times 1 cubed and we can ask if it has any other solutions, and if so, how do we find them? So another example of an equation with a Curve actually has a singularity as we will see later is something like y squared equals x squared plus x cubed so this is a solution 6 squared equals 3 squared plus 3 cubed or 24 squared equals 8 squared plus 8 cubed And again, we can ask can we find all integer solutions of this? So next example is the number 1729 is fairly famous for a reason I'll explain a little bit later And it can be written as a sum of two cubes in two different ways So you can ask are there other numbers that can be written as a sum of two cubes in two different ways So we're trying to solve W cubed plus x cubed equals y cubed plus c cubed And that there are some trivial solutions where w is equal to y or w is minus x or whatever So we we want to find non-trivial solutions of this For example, we could have 2 cubed plus 16 cubed equals 9 cubed Plus 15 cubed and again, we can ask how can we find solutions of this? Well one way to find solutions of all these equations is just a kind of stupid one We could just get a big computer and have it checking every possible case and This would sort of very slowly find solutions like this but what we want is a is a sort of lazier way of finding solutions with less effort and What I'm going to do is show how to find solutions of Diophantine equations by using geometry and the geometry makes it much easier to find solutions so let's look at the first example Pythagoras's equation and We can divide it by z squared. So if I put x equals x over z Y equals y over z and this equation becomes x squared plus y squared equals 1 And you can recognize this as the equation of a circle So what we're trying to do is to find points on the circle with rational coordinates So how can we find those systematically? Well, what you can do is there's an obvious point with rational coordinates, which is this one and I guess These four here, which just as coordinates minus 1 0 and What I'm going to do is I'm going to just draw a line through these two I'm just going to draw the straight line through these and This straight line has a slope It's called the slope t and the slope t is easy to figure out It's just this distance here divided by this distance here, which is just y over x plus 1 and Now we can work out the equation of this line So this line is just the line y equals x plus 1 times t Now you notice t is obviously rational if x and y are rational So we can say x y rational implies t is rational And we can ask the converse if t is rational or x and y rational And you can see the answer is yes without doing any calculations as follows Because this is circle has a degree 2 equation So if you try and find the intersection points, suppose you try and find the x coordinates of the intersection points of a line on the circle There'll be two intersection points which has a solution of some quadratic equation this quadratic equation is integer has rational coefficients and We know one of the roots is rational So the other route must be rational because the sum of the roots of a quadratic equation with in the rational coefficients is rational so We know before we do any calculations that if we Pick a rational t then x and y will be rational. Well, now let's actually calculate them So we have y equals x plus 1 times t and we also know x squared plus y squared equals 1 And if we substitute y into here, we find x squared plus x plus 1 t all squared Is equal to 1 and now this is a quadratic equation in x So it is two roots and one of the roots is minus one So it must factorizes as x plus 1 times something and you can work out what the something else is as t squared plus 1 times x plus t squared minus 1 equals 0 so here's the root and x equals minus 1 which corresponds to this point here and Here is the other route which corresponds to this point here and You see we can work at what x is we have x is equal to 1 minus t squared over 1 plus t squared and Y is equal to x plus 1 times t which is 2 t over 1 plus t squared So you can check by algebra now that if t is rational than x and y are rational But these are slightly complicated expressions and we could have found the x as we said We don't need to do this calculation to see that x and y are rational Anyway, we can now find lots and lots of solutions to Pythagoras equation just by choosing t For instance, if we choose t equals a half We find x equals three fifths and y equals four fifths Which gives us the equation three squared plus four squared equals five squared If we choose t equals it's just choose any random rational number three over eleven We find x is equal to 56 over 65 and y is equal to 33 over 65 giving us the solution 56 squared plus 33 squared equals 65 squared and you can obviously generate effortlessly generate enormous numbers of solutions to this just by picking a rational number t and Working out x and y from that So this actually gives essentially all solutions of this equation because solutions of this equation more or less correspond to rational numbers t There's one slight exception this point here doesn't correspond to a rational number t it corresponds to The point t equals infinity, but apart from that we get a one-to-one correspondence between solutions of this And rational numbers, so we've got a complete description of the solutions Um, there's so that's one geometric way of finding solutions There's actually a second geometric way of finding solutions of this So we again draw a circle so and This time Let's write down a solution Yes, we've got some point x1 y1 And let's just look at this angle theta And suppose we take another solution And let's call this angle phi So here we're going to have a solution x2 y2 and you can see this is equal to cosine phi sine phi and this one is of course equal to cosine theta sine theta And now what i'm going to do is i'm just going to add up theta and phi So now i'm going to have this angle here is going to be theta plus phi and the That the coordinates of this point will be cosine theta plus phi sine theta plus um phi And now um, we can use the formulas for the sum of two angles. So, um, this route here will just be cosine theta cosine phi minus sine theta sine phi And then we have cosine theta sine phi plus sine theta cosine phi And now what you notice from this is that if all these Numbers here are rational then these numbers here are also rational So what this means is if we've got a solution x1 y1 and a solution x2 y2 of Pythagoras's equation Then we can find a new solution as follows. So so we've got um x1 y1 With x1 squared plus y1 squared equals one and we've got x2 y2 x2 squared plus y2 squared equals one and now we take this point here, which is the point x1 x2 minus y1 y2 x1 y2 plus x2 y1 and if we call this x And this y then we see that x squared plus y squared will also be one So this gives a way of generating new solutions from old solutions using this formula Notice that algebraically this formula is a bit mysterious. It's not something you would Immediately think of but if you think of it geometrically, it's obvious what's going on. You're just adding up two angles And for example, we had a solution three fifths four fifths And we had another solution 56 over 65 33 over 65 And if we take this to be x1 y1 and this to be x2 y2 We can work out this equation here and it turns out to be 36 over 325 323 over 325 So we found another solution of Pythagoras's equation 36 squared plus 323 squared equals 325 squared and we can obviously carry on doing this and produce more and more larger and larger solutions And incidentally this shows that the solutions of Pythagoras's equation form a group Which means we can sort of There's a sort of way of adding up two solutions to get a third solution So here we take this solution and this solution and sort of add them together to get that solution Um Okay, now let's look at uh another Some equation So let's look at the equation y squared equals x squared plus x cubed So we have solution six squared equals three squared plus three cubed for example, and we want to find all solutions And we're going to do it geometrically So let's draw a graph of y squared equals x squared plus x cubed And it looks something like this You sketch it And you notice there's a sort of funny singular point here This is called a singularity So singularity just means a point where things go wrong. So here we've got two The curve is sort of crossing itself And we also notice there's one obvious solution here minus one zero as well as the solution zero zero Um, and now suppose you've got any other solution Say we've got a solution x naught y naught Well, what can we do with it? Well, what I'm going to do is I'm going to draw a line through this and another point And the most obvious other point to draw the line through is this singularity because there's obviously something rather special about it And let's draw the straight line through these two points And let's try and find the equation of this line. Well, it will have slope Is equal to y naught over x naught which is equal to t And this is a cubic equation So we expect every line every straight line will generally intersect in three points Because if we've got a if we set y equals a x was being substituted in we'll get a cubic equation for x So if x naught and y naught are rational then obviously t is rational So, um, let's just note that down. So x naught y naught rational implies t is rational And the question is if t is rational does this mean x naught and y naught are rational? and the answer is yes because If we fix t um, we have the equation y equals t x And if we substitute this in we will get a cubic equation for x And two of the roots are already rational because they're zero so the third root is also rational So let's just carry this out. So we substitute y equals t x and we get t x squared Equals x squared plus x cubed. So we have x cubed plus one minus T squared x squared equals zero and here we've got a cubic equation And you can see the roots are pretty obvious. There is a root zero and zero and the other root is T squared minus one and the two zero roots Um, we can see them there. They're where this line intersects the The orange curve in this double point and you see where it's very useful as a double point here because it means we get two Roots that are both zero Um, so now we can solve this equation We find x naught is equal to t squared minus one and y naught is equal to t times x naught, which is t times t squared minus one so Here's a way of finding solutions of this equation just by writing down rational numbers And of course, we want integer solutions So we might take t to be an integer. For instance, we could take t equals four and we get the solution x equals 15 y equals 60 and Similarly, you can generate as many solutions as you like um Well, that that cubic equation was particularly easy because it had a double point Um, what happens if you get a cubic equation without a double point? So let's try The example I mentioned earlier x cubed plus y cubed equals nine times z cubed And that's before let's just sketch this to see what's going on And it looks something like this. It's got a sort of asymptote at y equals minus x and it ends round like this So this is x cubed plus y cubed equals nine z cubed. Sorry. I equals nine. Sorry. I should have said I'm changing um Everything by dividing by z cubed so let's put x equals x over z and y equals y over z And then we find x cube plus y cubed equals nine And we know two points on this Because we've got the point one two And the point two one And how can we find other points on this? Well, well one idea is we could just draw a straight line through these two points Um, but that wouldn't be very interesting because the straight line Would just go like that and it would intersect this orange curve at infinity But it wouldn't have a third intersection point with finite coordinates. So what else can we do? Well, if we've got a a point on this curve, let's call its coordinates a b for the moment what we can do Is we can draw the tangent Um To the orange curve at this point so it looks something like this So here we've got a straight line And it intersects its orange curve in three points and it sort of Intersects it twice at this point here because it's a tangent vector. You see a tangent vector sort of Intersects a curve twice at one point So as usual, this means the third point of intersection will also have rational coordinates So we've got this very simple geometric idea if we Take a tangent vector to a solution and find the third point of intersection it will give us another solution So let's do this first of all we need to work out the coordinates of the tangent line well We work out the slope of the tangent line which is Minus a squared over b squared and then we can complete the tangent line as x minus a Plus b so it's beginning to look a bit messy And if we substitute this into this equation here We find we get x cubed plus minus a squared over b squared x minus a plus b All cubed equals nine and this is a cubic in x and It's got three roots So two of the roots are a Because it intersects This orange curve twice at a and there must be a third root Um Well, if you've got a cubic equation Something like a x cubed plus b x squared plus c x plus d equals naught You know the product of the roots is going to be minus d over a Um, so we can work out the third root just by saying The product of a and a in the third root is minus the constant term of this divided by the coefficient of x cubed and if you work this out it turns out to be minus a cubed plus b cubed all cubed Plus nine b to the six divided by b to the six minus eight six And then we have to divide by the two roots we thought of You can simplify this a bit because we know a cube plus b cube is equal to Nine for example, but anyway, so we have an explicit formula for the third root x x is now equal to this and similarly you can find y um So for example, so if we found one root we can now find another so let's take a b To be the root we already know about one two Well, this gives us the root minus 17 over 7 20 over 7 So we find minus 17 cubed plus 20 cubed equals nine times seven cubed So we found another solution of this Um, well, there's another way of finding solutions What we could do instead of taking a tangent line Is we can take any two solutions say this one and this one and draw the line through these So this will intersect in a third point And just as before if these two points are rational the coefficients of this green line will be rational So it's Intersection with the orange curve will again have rational coordinates um, if you do this the algebra becomes really a bit of a mess um So i'm not actually going to carry it out explicitly Um, but i'll just give an example. So if you take This point and this point two comma one And draw the tangent line through them and work at its intersection point. It turns out to minus two seven one over four three eight nine one nine over four three eight And this gives us the solution minus two seven one cubed plus nine one nine cubed equals nine times four three eight Cube which as you see would be a bit tiresome to find by trial and Using this geometric argument. It's quite easy to find solutions like this And obviously you can just keep on doing this Um, as long as your patient holds out every time you find a point you can take tangent line Every time you find two points you can draw a line through them So you can generate endless numbers of integer solutions to this equation in a in a fairly systematic way Um So the Um, there's actually a famous theorem related to this due to more dell Which says suppose you take any reasonable cubic equation All right, so for instance y was x cubed minus x might look something like this What you can do Is I guess it should go through the origin never mind you have to pretend Um, what you can do is whenever you've got three points on this On this cubic whenever you've got two points You can generate a third point by drawing a line through your first two points Um, this is called the chord tangent process for finding solutions Because you can either draw a tangent to a point you found or you can draw a chord through two points um, and It turns out that you actually get um Something called a group. So I suppose I've got two points p and q here Then I get a third point here and if I change its y coordinate to minus the y coordinate I get another point that I'm going to call p Uh sort of plus q. It's it's a funny sort of addition and it turns out That this funny addition of points behaves obeys the usual rules of algebra So p plus q is obviously equal to q plus p Because the lines through p and q is the same as the lines through q and p um Slightly more difficult fact is that p plus q plus r is equal to p plus q plus r This is really a total mess to verify because the formula for the coordinates of p plus q um It are really rather complicated and trying to calculate this explicitly is really rather a headache Um, there is actually an easier way to prove this without doing explicit calculations, but whatever so um What we get is a sort of funny addition of points which obeys the usual rules of algebra And punk hurry ask this question. Can you find a finite number of rational points on any cubic? So that all rational points can be obtained from this finite number by by using this operation And mordell showed the answer is yes He showed the rational points Are finitely generated In other words, you can find a finite number of rational points Such that all the other rational points are generated from these by the chord tangent process um, so um So now we get so far. We've been looking at curves now. I'm going to look at an example of a surface. So this is Related to remanigans famous number So there's a famous mathematical anecdote about this. Um, this appears in hardy's book on Remanigans, so I have here a copy of hardy's book And on page 12 he has he has this anecdote that I'll just read out So hardy says I remember to go going to see him once when he was lying ill in chutney So he's he's talking about remanagement, of course I had written in taxi cab number 1729 and remarked that the number seemed to be a rather dull one and that I hoped it was not an unfavorable omen No, he replied it is a very interesting number It is the smallest number expressible as a sum of two cubes in two different ways I asked him naturally whether he would tell me the solution of the corresponding problem for fourth powers And he replied after a moment's thought that he knew no obvious examples and suppose the first such number must be Very large So as a result of this story this this particular number 1729 has become Really famous. It's even got its own wikipedia article if you want to read about it um so What i'm going to do is show how to find Other examples of numbers with this property So we want to solve a cube plus b cubed Equals c cube plus d cube and obviously we want Non-trivial solutions where a isn't equal to c or d and it's not equal to minus b and so on You can rearrange this by changing the sign of c and d so Uh, we so it becomes a bit more symmetrical It looks like that and now we can just divide by d cube. So let's put x equals a over d y equals b over d z equals c over d So we get x cubed plus y cubed plus c cubed equals one Um, so this is a bit like the earlier equation. We had x cube plus y cube equals nine except now We've got three variables and this is now the equation of a surface in Real three-dimensional space and we're trying to find points on the surface with rational coordinates um and now We can do a similar trick that we did with cubic curves. Let's take two points p and q And draw a line through them And this line will usually intersect this cubic surface in another point. So r is the intersection point of this line and the surface x cubed plus y cubed plus c cubed equals one assuming it line intersects this occasionally the line would just be asymptotic to it, but we'll ignore that problem And now just as before if p and q have rational coordinates Then the equation of this line will have rational coordinates So the third intersection point with this surface will also have rational coordinates So this will be a rational point Okay, well, let's try this first of all We notice this equation has lots of obvious solutions like it's got x cube plus minus x cubed plus one cube equals one cubed and it's got um y cubed plus one cubed plus minus y cubed equals one cubed and One cubed plus c cubed plus minus c cubed equals one cubed. So these are the um obvious not very interesting solutions Um and we can ask what happens if we take two of these boring solutions and do this process We draw the line through them and take the third intersection point and the answer is you get a third uninteresting solution So if you take a point on with satisfying this and a point satisfying this and carry out this process We just get a point here. So that's not terribly interesting However, we've got um a more interesting solution minus nine cubed plus minus 10 cubed plus 12 cubed um plus one equals zero um, so um I guess it should be a minus one there never mind um, so Uh, what happens if we do this? Well, let's just see. So let's take the point p to be one minus one minus one And we're going to let this be Correspond to a rational number t being naught and at t equals one We're going to let it be the point q which is equal to minus nine minus 10 12 And now we're going to draw the line through these points and it will be the set of points one minus 10 t Minus one minus nine t minus one plus 13 t so this is a line through p and q because you can see it's Line parameterized by t and it goes through p and q when t is not all one. So this is going to be our third point r And now we we want the sum of these three numbers cubed You want x cubed plus y cubed plus c cubed plus one to be equal to zero So we get one minus 10 t cubed plus minus one minus nine t cubed plus minus one plus 13 t cubed equals one So plus one Zero keep getting that the wrong way around um, and you can see that it has three roots It has roots t equals zero Or t equals one and the third root Is going to be rational and you can work it out explicitly by doing some calculation It turns out to be minus one over 26 in this case So we now find our point r is equal to 36 over 26 minus 17 over 26 minus 39 over 26 Um, so we get a solution 36 cubed plus 26 cubed Equals 17 cubed plus 39 cubed. So we've got another solution to ramanogen's equation um and um Um, obviously you can keep generating more and more solutions by just drawing a line Through any two solutions you've got um incident you can ask If you get a group like this and the answer is you don't because if the points p and q are equal You think you would take the tangent line to p and q But the trouble is if you've got a point of a surface it doesn't have a tangent line It has a tangent plane So you don't really know which Line tangent to it to take so you can't really define An addition operation on points because this breaks down if the two points are the same um so um in general what happens with playing curves um is If you've got a plane if you've got a curve then it might have degree two And if you've got a plane curve of degree two It acts rather like the equation x squared plus y squared equals one if we can find one rational point on it Then we can find all the other rational points very easily just by drawing a line through both points If we've got a degree three curve Such as x cubed plus y cubed equals nine Then um, this is an example of something called an elliptic curve Unless it happens to be singular such as the curve y squared equals x squared plus x cubed Um, so singular elliptic curves with singularities behave a bit like degree two curves elliptic curves without singularities behave like the degree three curve we had That's you more del showed you can find a finite number of points that generate all of them Um, what about degree curves of degree greater than three? so If you have something like this um Then um curves of degree greater than three Generally only have a finite number of rational points on them This is a very deep theorem proved by faultings a few decades ago. It was originally conjectured by moor del So anyway, we we turns out that curves split up into three different groups There are those for which the rational points are really easy to find An infinite number usually Um, then we get elliptic curves Where the rational points are at least finitely generated And then we get all the other curves where there are only a finite number of rational points and they seem to be very hard to find Um, these differ by something called the genus. We say these curves have genus zero these curves of genus one These curves of genus greater than one And the genus turns out to be what happens if you take the curve over the complex numbers Which gives you a surface and then the surface has a number of Sphere with a number of handles and the number of handles is the genus So we've got this really weird relation between the topology of a complex curve and the complexity of the number of points on it Okay, I think I'll just finish by giving a suggestion for further reading if you want to find out more about this Um, there's an introductory book called undergraduate algebraic geometry by miles reed and the early chapter on it explains um This method of generating um Rational points on cubics. So here chapter two is on cubics and the group law And he explains this method of generation new points from old points in more more detail Okay, I think that's all for this lecture