Wildberger solves the twin prime conjecture!!
Insights into Mathematics
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4,047 views Apr 29, 2024
Does it really make sense to "go to infinity?", or to "take the limit of an infinite process?", or to "calculate an infinite sum?" Well in this lecture, we depart from our usually rigorous approach to pure mathematics, and accept the standard orthodoxy that we ARE allowed to do all these things, and then show that this leads to an essential collapse in number theory. Almost all the major unsolved problems can more or less straightforwardly be dealt with by simply applying this super power to make the crucial calculations to determine truth values of propositions involving an unbounded number of cases. We initiate this exciting new chapter in number theory by solving the Twin Primes Conjecture: not by any sophisticated argument, but just by making a pedestrian computation: at least pedestrian by the current standards of modern analysis and number theory, which have no qualms in "computing cos(7) or zeta(5)" or a host of other "transcendental function values." In fact the heart of the matter is simply that the series 1/2+1/4+1/8+... ostensibly converges to 1. Hopefully this should be a long overdue wake-up call, at least to students and young people entering the field. Are we not obliged to think clearly about what we are actually talking about in pure mathematics??? There is a much richer, more careful and beautiful mathematics out there, where everything actually makes 100% sense, and where there are multitudes of explicit examples to support our favorite notions. Our computers are soon going to be all over this terrain: but it's not too late to change gears and join the future. And in particular, start doing number theory in a more careful, interesting and ultimately rewarding fashion. A big thanks to all my Patreon supporters!
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[Music]
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Hello, everyone. I'm Norman Wildberger. Today, we're going to look at the sociology of going
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to infinity, and we're going to apply some basic logical reasoning to, well, largely kill number
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theory. We're going to do that by showing that if you accept the rather unwarranted assumptions,
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from my point of view, of modern analysis and number theory, then you can basically solve
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almost all famous number theoretical problems at the drop of a hat. I'll exhibit this in the
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context of one particular problem, but the argument is actually quite general, and it's
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an interesting exercise for undergraduates or graduate students to simply convert the argument
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that I'm showing you today to solve basically any other important open number theory problem.
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So, let's look at the twin prime conjecture, which asks whether there's an infinite number
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of twin primes, that is, primes which are exactly two apart. And what I'm going to suppose is that
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I am in possession of what all my mathematical peers assume they are in possession of, namely an
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ability to do an infinite number of things. So, for example, cosine of seven—that's an infinite
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series, and to assert that there really is such a number is essentially to assert that you're
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able to do this infinite number of arithmetical operations and get a bona fide result at the end
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of it. Not just cosine of seven, but say, zeta of five—the zeta function: 1/15 + 1/25 + 1/35,
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etc., to infinity. Right, it's a staple of number theory. That's an important function
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that tells us a lot about prime numbers, and it really does make sense; there really is such a
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thing as zeta of five, which is what you get when you add up all these infinite number of
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things. So that's the position I'm going to take. That's not my ordinary position,
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but just for the sake of this particular talk, I'm going to assume that I have these superman
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powers along with my fellow mathematicians, and let's solve the twin prime conjecture, shall we?
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So, without even using the blackboard, I'm just going to explain an algorithm. We're going to
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compute a number, which determines the truth value of this proposition,
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that there are an infinite number of twin primes. So just a little bit of programming
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is all that we'll need. We're going to start with a total. The total is currently zero,
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so there's a box called total, we're putting zero in there, and we're going to add stuff to
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that. And what's in that total box at the end is going to be the thing that really interests us.
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We're going to start with the number one, and we're going to work our way through the natural
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numbers. Actually, since we're looking for twin primes, it'll suffice to just
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go through the odd numbers. And we're going to be searching for when we get a twin prime pair. So,
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let's start with one. Well, one is not a prime, generally considered, so one and three—that's not
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really a twin prime pair, so that doesn't give us anything. The next one, the next odd number,
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is three, and then three and the next one is five. Ah, that's a twin prime. We found
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our first twin prime pair. So over here we have a counter, which we're going to set to
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one because we found the first twin prime pair. So our twin prime pair counter is equal to one,
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and consequently, I'm going to change the total. I'm going to add to the total 1/2 to
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the twin prime counter, so 1/21. So now the total has a total of a half in it. Excellent.
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Now we're going to augment our odd numbers to five and ask, hmm, five and seven—is that a twin prime?
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It's easy to calculate that, right? If we're able to do an infinite number of things, we can
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certainly calculate whether any given number is prime. That's done in a snap, and five and seven
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are both prime, so hey, there's another twin prime pair. So our twin prime pair counter goes up from
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one to two, and since it's just been augmented to two, it means we have to augment our total.
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We're going to add to our current total 1/22. So now in total we have 1/2 plus 1/4, which is 3/4.
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The next one is seven and nine. That's not a twin prime pair. Nine and eleven, that's not a
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twin prime pair. Eleven and thirteen—ah, that's a twin prime pair. Okay, no twin prime pair,
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so our twin prime pair counter goes up from two to three, and we're going to add to our
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total 1/23. And we're going to carry on like this. So the total—what's going
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to happen to the total? At each stage, we're adding the next reciprocal, 1/2k,
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if at all. If something is being added, it's 1/2k, where k is one more than the previous one.
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And such an addition happens every time we find a new twin prime pair.
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And so now, ladies and gentlemen, we are going to go to infinity,
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as my analyst friends do all the time, as my number theoretical friends do all the time,
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as my algebraic geometer colleagues do all the time. We just go to infinity. We pretend—no,
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sorry, we assume—that we can just carry this computation out to the end. Are
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we all in agreement that we can do that? Yes, we're all nodding. We have beatific
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smiles. We are pure mathematicians. We stride across the universe of thought.
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So we have gone to the end of this algorithm. Let's have a look: what's in this total? Well,
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if there's only a finite number of twin primes, then what's in that total is
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going to be a rational number which is strictly less than one; it's between zero and one. So,
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the form 1/2 plus 1/4 plus 1/8 plus 1/16, etc., up to some 1/2k. However, if there are an infinite
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number of twin primes, then we will have gone through the entire series: 1/2 plus
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1/4 plus 1/8 plus 1/16, etc., to infinity. And we all know what that sums to: it sums to one.
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So, in our final step, what we're going to do is we're going to take that number total
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and we're going to take the greatest integer part of it. That means that if it's an integer,
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it stays an integer. If it's something like 2.7, the greatest integer part is two. If it's 0.89,
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the greatest integer part is zero. So we take the greatest integer part of this total,
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and we're either going to get zero or one. That's the result of our calculation. And in principle,
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it's not really so different from a calculation like cosine of seven or zeta of five. It involves
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an infinite number of arithmetical steps, each of which is reasonably completely clear.
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It's just that as we go further and further, the numbers involved get bigger and bigger.
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In practice, of course, eventually they get so big that they don't fit into the universe,
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but that doesn't bother our analyst friends. They just pretend that you can still do these
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arithmetical infinite computations, and we're going to follow that route.
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So this total that I've obtained is a completely legitimate object. It is either zero or one,
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whatever is in there, and that is the truth value of the proposition that there are an
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infinite number of twin primes. So I have solved that conjecture here for you today completely
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computationally. It's a computation that we have made in the same way that one calculates
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or talks about cosine of seven or zeta of five or any number of other such transcendental
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functions in mathematics, without even a qualm, without even the slightest hint of embarrassment.
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So, of course, this elevates me to a very high status in the world of number theory now,
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especially once a little bit of thought is applied. You see that basically the
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same argument, or a very similar one, can be used to solve pretty
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well every famous number theoretical problem. And so that's a great exercise:
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go ahead and solve some famous problems. You can become famous this afternoon. Just write
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out exactly the parallel kind of argument for your favorite number theory problem,
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and then you will be credited forever after with having solved that problem by pure thought.
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In my next video, I'm actually going to apply the same reasoning to the most famous unsolved
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problem in mathematics, and let's see if we can crack the Riemann hypothesis.
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I'm Norman Wildberger. Thanks for listening.