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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.

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    $\begingroup$ 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? $\endgroup$
    – Arthur
    Jun 4, 2021 at 9:20
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    $\begingroup$ It kills me how I have two answers AND a comment on this question, but NOBODY has upvoted the question. So people think it's worthy of an answer or comment, but not worthy to be upvoted. How does that work? $\endgroup$ Jun 4, 2021 at 19:53
  • $\begingroup$ The ways of the upvote fairies (and the downvote fairies as well) are ineffable, unfortunately. $\endgroup$
    – Arthur
    Jun 4, 2021 at 20:28
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    $\begingroup$ @JimmyG. What are you going to do with the upvotes or downvotes as you say that you don't care whether the number of twin primes is finite or infinite. $\endgroup$ Jun 4, 2021 at 23:16
  • $\begingroup$ @JimmyG. HAHAHAHAHAAAAAA!!! I know. Tough crowd. $\endgroup$
    – JustKevin
    Oct 25, 2021 at 20:06

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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?

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  • $\begingroup$ Not to mention have consequence on other conjectures. You can tie it in with strong Goldbach, and probably a few more. $\endgroup$ Jun 4, 2021 at 19:30
  • $\begingroup$ 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. $\endgroup$
    – JustKevin
    Oct 25, 2021 at 20:03
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    $\begingroup$ @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". $\endgroup$ Oct 25, 2021 at 20:36
  • $\begingroup$ 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. $\endgroup$
    – JustKevin
    Oct 25, 2021 at 20:57

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