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The twin-prime conjecture (also known as Polignac’s conjecture, 1846) states that there are infinitely many twin primes (pairs of primes that differ by 2; for example, 3 and 5, 5 and 7, 11 and 13, and 17 and 19 are twin primes).

We know  that as numbers get larger, primes become less frequent and twin primes rarer still.

In 1919, Viggo Brun showed that the sum of the reciprocals of the twin primes converges to a sum, known as Brun’s constant (approximately 1.90216054).

In 2003, the next big breakthrough occurred. Daniel Goldston and Cem Yıldırım published a paper, “Small Gaps Between Primes,” that established the existence of an infinite number of prime pairs within a small difference. Yitang Zhang showed in 2013 (Bounded gaps between primes) that, without any assumptions, there were an infinite number of pairs of primes differing by at most 70 million (this bound was improved to 246 in 2014).

Even though it is clear what this conjecture asserts, I cannot grasp yet why it is so relevant as to be considered one of the biggest open problems in number theory. Which are its implications? What are the problems that will be solved with the same possible solutions?

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    $\begingroup$ I think the consensus is that this problem is not important because of its potential applications, but because of how it reveals how limited our understanding of the distribution of prime numbers is. $\endgroup$ Feb 17 at 17:28
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    $\begingroup$ Goldston and Yildirim don't have a paper entitled "Small gaps between primes". You forgot about the third co-author, János Pintz. More precisely, Goldston, Yildirim, and Pintz published a paper under this title in the proceedings of the 2014 ICM. But the original ground-breaking results appeared as Goldston-Pintz-Yildirim: Primes in tuples I-IV". Note also that Polignac's conjecture is a generalization of the twin prime conjecture: every even number occurs as a prime gap infinitely often. $\endgroup$
    – GH from MO
    Feb 17 at 18:20
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    $\begingroup$ @SamHopkins: There are some connections. For example, Heath-Brown proved that if suitable Landau-Siegel zeros exist, then the twin prime conjecture is true. An other example is the ternary Goldbach problem that was originally proved under the RH. See also the work of Harald Helfgott (fellow MOer) in this regard - how the zeros of Dirichlet $L$-functions are relevant. $\endgroup$
    – GH from MO
    Feb 17 at 18:29
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    $\begingroup$ To expand on what @GHfromMO wrote, Hardy and Littlewood showed nearly 100 years ago that GRH for Dirichlet $L$-functions implies for each $r \geq 3$ an asymptotic formula (on the order of $n^{r-1}/(\log n)^r$, as $n \rightarrow \infty$ through integers of the same parity as $r$) for the number of representations of large $n$ as a sum of $r$ primes when $n \equiv r \bmod 2$. They were unable to extend their method down to the case $r = 2$ (classical Goldbach conjecture). $\endgroup$
    – KConrad
    Feb 17 at 23:53
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    $\begingroup$ You could ask the same question about other elementary-sounding results, such as Fermat's Last Theorem. Whether or not it was true was not known to have any profound consequences until the mid-1980s when it was shown that a counterexample to FLT would lead to counterexamples to other results that mathematicians had much more professional interest in (modularity of elliptic curves over $\mathbf Q$). The methods created to settle FLT then inspired new ideas that led to the solution of other famous problems (in number theory) like the Sato-Tate conjecture. $\endgroup$
    – KConrad
    Feb 18 at 0:01

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As far as I know, the twin primes conjecture doesn't have applications.

It is considered interesting because it is an extreme example of the kind of simple to state, hard to solve problems that are favored in number theory. It's also one of the older open problems in number theory, though surely not the very oldest. Finally, it's considered a good test of the strength of our methods.

Any approach that solves the twin primes conjecture will likely have applications to other problems on the distribution of prime numbers. In particular, it's reasonable to guess that it would lead to a solution to Goldbach's conjecture, at least for sufficiently large n. It may involve progress on Chowla's conjecture and the randomness of the Mobius function. It could also lead to progress on pair correlation of the zeroes of the Riemann zeta function.

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    $\begingroup$ It's bound to have at least some applications - it has to be a natural, useful assumption in some contexts (not necessarily in number theory). On the whole, of course you are right. $\endgroup$ Feb 17 at 17:38
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    $\begingroup$ While I don't know of any direct applications of the infinitude of twin primes, the closely related conjecture that there are infinitely many Sophie Germain primes (or equivalently), infinitely many safe primes has some implications in cryptography, I think. en.wikipedia.org/wiki/Safe_and_Sophie_Germain_primes $\endgroup$
    – Terry Tao
    Feb 17 at 19:39
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    $\begingroup$ Sophie Germain primes do have applications in cryptography. It's less clear that proving that there are infinitely many Sophie Germain primes would have any impact on cryptography. $\endgroup$ Feb 19 at 0:41
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Speaking from the point of view of someone interested in prime number theory, the fact that the statement "there are infinitely many pairs of primes $p,q$ with $|p-q| \leq 6$" is provable conditionally (on the extended Elliott-Halberstam conjecture) but the twin-prime conjecture is NOT provable even if we throw everything we have at it, both proven and conditional, is quite remarkable. The key is that prime gaps of size at most $6$ do not require us to break the parity barrier, while twin primes does.

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    $\begingroup$ This is not quite right. The result that was proven conditionally on generalized EH was that there are infinitely many primes $p,q$ with $|p-q| \leq 6$, or that there are infinitely many pairs of primes $p,q$ with $|p-q|=n$ for at least one of $n=2,n=4,n=6$. $\endgroup$
    – Will Sawin
    Feb 18 at 4:48
  • $\begingroup$ @WillSawin you are right of course, and I even thought of putting in the inequality sign, but somehow my fingers slipped and the equality sign got there. I will fix it $\endgroup$ Feb 18 at 17:51

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