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June 23

Strengthening of the twin prime conjecture

Here, David Eppstein (talk · contribs) removed an unreliably sourced attempt at a proof of the conjecture. The given proof is incomplete (specifically, the bounds on the number of remaining twin prime pairs after running the sieve of Eratosthenes need justification), but the "theorem" (which I will state here as a conjecture, given the lack of actual proof) claimed seems to merit more discussion.

Conjecture Let a and b be a twin prime pair (that is, a and b are both prime and $b=a+2$ ). Then there exist at least two distinct twin prime pairs in the interval $(a^{2},b^{2}]$ .

Thus far, I can't find any counterexample; it would obviously imply the twin prime conjecture, but is a strictly stronger statement, so it could have counterexamples without disproving the twin prime conjecture. What known results would it violate?--Jasper Deng (talk) 07:57, 23 June 2019 (UTC)[reply]

According to Wesolowski's comment in OEIS:A091592, there is at least one twin prime between n^2 and (n+1)^2 for n from 123 to 10^7. It follows from this and the lack of small counter examples that the conjecture holds up to 10^7. The important conjecture about the number of twin primes is the asymptotic number in Twin prime#First Hardy–Littlewood conjecture. Anyone can make a conjecture about specific values, and lots of people do. They are usually very weak compared to Hardy–Littlewood, like this one. The expected number of twin primes between any n² and (n+1)² tends to infinite. Since there is no small counter example, the conjecture seems very likely to be true. I think this is a typical way to produce number theoretic conjectures:

Make a guess which is almost certainly true for large values because it's weak compared to the expectation.
Test for small counter examples.
If no counter example is found then call it a conjecture. Otherwise, adjust the guess until small counter examples are eliminated – often by just saying "for numbers above x", where x was the largest found counter example.

I think most professional mathematicians have low interest in such conjectures but they are popular in many amateurs. Sun Zhiwei is a professional who makes loads of such conjectures, some of them about twin primes. PrimeHunter (talk) 11:09, 23 June 2019 (UTC)[reply]

@PrimeHunter: I had hoped that weaker conjectures of this sort might be studied as a possible proof route for the twin prime conjecture, as this crank had thought he had here.--Jasper Deng (talk) 11:29, 23 June 2019 (UTC)[reply]

A strong form of Cramér's conjecture suggests that the longest gaps between twin primes should be

O(\log ^{3}x)

, much smaller than would be needed for this weaker conjecture to be true. But if we can't prove Cramér's conjecture itself and we can't prove the twin prime conjecture, why should we hope to be able to prove a combination of both? Anyway, the supposed source is more than incomplete; it has nothing resembling a proof in it. It is a mere announcement of a claim without support. I don't think it's worth the time we've already spent discussing it here. —David Eppstein (talk) 06:22, 24 June 2019 (UTC)[reply]

In general it's a good thing to have a few famous unsolved problems to point to so that people don't get the impression that everything in math was decided hundreds of years ago and there is nothing new to be learned. But it has the unfortunate side effect that people think mathematicians spend most of their time thinking about these specific problems. Another unfortunate side effect is that it attracts cranks who get the idea that the path to fame and fortune is to solve one of these problems. (I've said it before, but if you're after fame and fortune then you're much better off learning the guitar instead of math.) It's often not understood how rare it is for a famous unsolved problem to be solved; the Four color theorem and Fermat's Last Theorem both fell in my lifetime and I'm pretty sure I'm not due for a third. Another factor that's not often appreciated is that there usually has to be a fundamental change in techniques before a solution becomes possible; for Four color it was computers and for Fermat it took some technical advances in the theory of elliptic curves. The upshot is that the public has an inordinate interest in a few mathematical problems which actual mathematicians have very little if any interest in pursuing. In other words, even though the twin primes, Goldbach and Colatz conjectures may be popular on YouTube, posting about them here will at best result in a rousing 'Meh.' --RDBury (talk) 13:38, 24 June 2019 (UTC)[reply]

I'm pretty sure I'm not due for a third. Nuts to Grigori Perelman, then? :p --JBL (talk) 18:21, 24 June 2019 (UTC)[reply]

The Poincaré conjecture, significant yes, but what I had in mind were problems known to the general public; the kind of thing that YouTubers try to explain it to people with high school level math, or see mentioned in an episode of Star Trek. (I realize you're teasing a bit but you do have a point. The Kepler conjecture is also a candidate.) Now I'm wondering if this was 1919 instead of 2019 what would be on the list. Transcendence of π maybe. --RDBury (talk) 20:36, 24 June 2019 (UTC)[reply]

On π, Ferdinand von Lindemann had proved it transcendental in 1882.John Z (talk) 08:46, 27 June 2019 (UTC)[reply]

Riemann hypothesis may be proven in the near future. Count Iblis (talk) 11:28, 25 June 2019 (UTC)[reply]