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From what I understand, the main premise of the twin prime conjecture is "Are there an infinite number of twin primes?" And twin primes are prime numbers that are separated by two. Examples include: $(3,5), (5,7), (11,13), (17,19)... (793517,793519), (793787,793789), (793841,793843)... (2924351,2924353), (2924567,2924569), (2924921,2924923)... (7120187,7120189), (7120277,7120279)... (12382691,12382693), (12382691,12382693)... (16148159,16148161)... (17355509,17355511)... (18409199,18409201)$, etc.

If I have something wrong, please tell me. If I have it correct, please explain to me why this matters. What I mean by why it matters, is what effect will it have on the real world. Usually when I hear of the practicality of prime numbers, it is in reference to cryptography. So, if there are an infinite number of twin primes, does this mean good for white hats, bad for black hats? And what if there are not an infinite number of primes and we learn them all. What implications will that have in the real world. Does the importance of the twin prime conjecture go beyond cryptography? If so, please explain. Thank you.

Ritam_Dasgupta
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Curious Layman
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    It's a very simple question that's been asked for two and a half millennium, and still no one knows the answer. Wouldn't you find that intriguing? – Arthur Jun 04 '21 at 09:20
  • "Why it matters" are "what effect it will have on the real world" are two very different things. If you disagree, fine, but most mathematicians wouldn't, and they're the ones studying and doing number theory. – anomaly Apr 17 '25 at 15:36
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    "If I have it correct, please explain to me why this matters." Who said it did? "What I mean by why it matters, is what effect will it have on the real world" It will have no effect. So what? The major effect is that it will be interesting to know why it is true or false. And if people are allowed to have the opinion "That doesn't matter" aren't I allowed to have the opinion "It doesn't matter than it doesn't matter"? – fleablood Apr 17 '25 at 15:37

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A solution of the twin prime conjecture most likely will show completely new ideas and techniques in the area of analytic number theory. The result itself will not really matter for applications, but the methods might be very helpful for later applications.

As for real-life applications of prime numbers in general, there has been said more than enough on this site, compare for example this post (and others):

Real-world applications of prime numbers?

Dietrich Burde
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    Not to mention have consequence on other conjectures. You can tie it in with strong Goldbach, and probably a few more. – Roddy MacPhee Jun 04 '21 at 19:30
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    I hate to be a skeptic, but I'm not so sure about that, because a lot of the framework is already there and has already been there for over a century. I don't see how such an old problem could possibly show "completely new" ideas and techniques. It'd be more realistic to admit that "a" proof might not be quite as mathematically impressive as some have been led to expect. – JustKevin Oct 25 '21 at 20:03
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    @JustKevin The same skepticism was said about "such an old problem" by Fermat. But indeed, so many new techniques and ideas has arisen from it. I do not claim that this will be the same for the twin prime conjecture, but I also don't see why it should be "not quite impressive". – Dietrich Burde Oct 25 '21 at 20:36
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    Twin Primes and Goldbach is more of an exercise in "exhaustive triviality" where there are more elements of hashing, I guess. Looking at Mersenne and Fermat we had to do some figuring as far as what the "generators" of multiplicative sets are, and that, as I understand it, is a much harder problem to grasp than, well, merely setting up a total product between two arithmetic progressions and doing some figuring about how the combinatorics describe the asymptotic. Both Bombieri-Vinogradov and Hardy-Littlewood are old news. – JustKevin Oct 25 '21 at 20:57