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There are many prime representing functions.

For example, $\lfloor A^{3^n} \rfloor$ is prime representing function because for all positive integers n ,it generates a different prime number. Here $A$ is approximately $1.306377...$ and $\lfloor . \rfloor$ denotes the floor function.

Likewise, is there any function that generates twin primes for all positive integer n?

Question Number 2
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    Sure: $f(n) = 11$ is a twin prime for all integers $n$. – Martin R Nov 13 '23 at 14:32
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    Nice joke :) Then I should fix my definition. – Question Number 2 Nov 13 '23 at 14:36
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    Re your edit: The existence of a function generating distinct primes for all positive integers would imply the existence of infinitely many twin primes, which is an unsolved problem for almost 200 years. – Martin R Nov 13 '23 at 14:44
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    Wow, I never thought of it this way. So this question is useless because if we can find this kind of function, we also prove the conjecture. – Question Number 2 Nov 13 '23 at 14:51
  • I think that there is some recent result that states that $\liminf (p_{n+1}-p_n)$ is finite, but this is still a bit far from the twin primes' conjecture. Can't remember the author now, maybe he is from Japan? I have not the slightest idea if the proof involves a generating function, though. – ajotatxe Nov 13 '23 at 14:51
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    @ajotatxe: Do you mean this (from https://en.wikipedia.org/wiki/Twin_prime#Twin_prime_conjecture)? – “On 17 April 2013, Yitang Zhang announced a proof that for some integer N that is less than 70 million, there are infinitely many pairs of primes that differ by N.” – Martin R Nov 13 '23 at 14:56
  • @MartinR Yes, it seems so. And he's from China. But I was thinking about Terence Tao, who is from Australia (though his ethnecity is Chinese, too). – ajotatxe Nov 13 '23 at 14:59
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    @MartinR Also, I just realized that your first comment was not joke. – Question Number 2 Nov 13 '23 at 15:15
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    The best unconditionally proven result is that infinite many prime gaps do not exceed $246$ , but that does not mean that there is a constant such in the case of the primes doing the job. – Peter Nov 14 '23 at 06:27
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    See my second-to-latest answer on MSE. I cover almost exactly what you're asking for including full derivation: https://math.stackexchange.com/a/4814449/26327

    Requirements: modular arithmetic, floor, primorial, sieve-of-Eratosthenes. This is more of a twin-prime counting formula rather than a spit out some twin primes formula. But it's a good starting point. It is actually a generalization of the inclusion-exclusion formula for $\pi(x)$ (usual prime counter) as seen on the prime-counting wikipedia page under algorithms for computing.

     – Daniel Donnelly Dec 12 '23 at 01:40
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    @DanielDonnelly Your approach is correct. After all, this is a combinatorics problem. To deal with this type of problem, techniques from sieve theory are normally used, but despite the advances, the harder is to obtain a proof for a given problem (such as the twin prime conjecture). The issue of definition is the easiest part. – Yapet G. Dec 12 '23 at 02:25

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