Simultaneous proofs

Two open problems about the distribution of the primes have been solved within about 24 hours of each other, namely the ternary Goldbach conjecture and a weakened form of the twin prime conjecture. Let’s look at these in approximate chronological order:

Ternary Goldbach conjecture

This states that every odd integer n \geq 5 can be expressed as the sum of three primes. Hardy and Littlewood first proved this for sufficiently large integers, conditional on the truth of the Riemann Hypothesis. Later, in 1937, Vinogradov proved the result for sufficiently large numbers without requiring the Riemann Hypothesis; this became known as Vinogradov’s theorem.

An lower bound was established, which was gradually reduced to e^{3100} in 2002, still ahead of the computer searches (which have only probed up to about 10^{18}). Very recently, this was further reduced to 5, as desired; see the paper by Helfgott.

We can corollary-snipe† at this point, and state that this trivially implies that every integer n \geq 8 can be expressed as the sum of four primes. The full version of Goldbach’s conjecture is equivalent to every integer n \geq 6 being expressible as the sum of three primes.

Twin primes conjecture

Twin primes are primes which differ from their closest prime neighbours by 2. For instance, {5, 7}, {59, 61} and {65516468355 \times 2^{333333} - 1, 1 + 65516468355 \times 2^{333333} } are examples of pairs of twin primes. It has been conjectured that there are infinitely many such pairs, although no proof is known (there was a purported proof in 2004, but it was fallacious).

It has now been proved that infinitely many pairs of primes have a distance of at most 70000000. Again, this can be corollary-sniped to deduce that there exists some N \leq 70000000 such that infinitely many pairs of primes are separated by precisely N. No explicit values of N are known, and the original conjecture states that 2 has this property.

Conclusion and footnotes

Together, these results provide another piece of evidence for the following conjecture:

There are infinitely many pairs of exciting proofs published within 70000000 milliseconds of each other.

† Corollary-sniping is a rather impolite and dishonourable practice in which one jumps on a big theorem proved by someone else, and proves one or more corollaries using it. For instance, if someone suddenly exclaimed “Hence Fermat’s Last Theorem!” just as Andrew Wiles proved the necessary cases of the Tanayama-Shimura conjecture, that would have been an epic case of corollary-sniping (if that happened, hopefully the prize would still have been awarded to Wiles). An actual instance was when Xia’s proof that particles can be projected to infinity in finite time under Newtonian gravity was famously corollary-sniped to deduce that the n-body problem is undecidable.

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17 Responses to Simultaneous proofs

  1. wojowu says:

    Actually, 5 can’t be represented as sum of 3 primes. But it can be with at most 3 primes

    • apgoucher says:

      Oh, yes, by 5 I meant 7. By the way, is your surname Nadara? If so, some of my friends are acquainted with you.

      • wojowu says:

        No, I’m not Nadara, and I don’t think I know any of your friends, because I live in middle Poland. But from small-world phenomenon we aren’t that far from knowing each other 😉
        Also, from what I know, Goldbach’s weak conjectures says about sum of 3 odd primes. I can’t see obvious equivalence, because, we can have number of form p+2+2 which may not be sum of odd primes below p. But I think Helfgott’s result estabilishes both cases.

  2. Ross Presser says:

    If you assume that the universe as we know it will someday die (by heat death or by Big Rip or by Big Crunch) and that there will never be an infinite number of mathematicians, your conjecture is false.

    • Andrew Carlotti says:

      I think you may also be making the assumption that there is a minimum length of time for which mathematicians exist, and a maximum rate at which they can produce exciting proofs.

      • apgoucher says:

        You’re also assuming that the universe is finite. The observable universe certainly is, but little is known about the large-scale structure of the universe (for instance, we don’t even know whether it is simply-connected).

        • Andrew Carlotti says:

          He’s already accounted for that (“will someday die” and “never be an infinite number of mathematicians”), although makes the implicit assumption that time works in a simple manner.

          • apgoucher says:

            In that case, the only possible loophole is if a mathematician completes a supertask (infinitely many theorems in a finite time).

          • Andrew Carlotti says:

            I’m afraid we’ve already covered that possibility too (“a maximum rate at which they can produce exciting proofs”).

          • apgoucher says:

            Yes, you did in your later comment, but not in the original poster’s comment. I think we have all bases covered.

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  4. Squib says:

    More corollary-sniping, except this one actually happened. At least the author recognizes that that’s what’s going on.

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