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(Above, $ℙ≔\{\text{all primes}_ℤ\}$.)

Are there any analytic functions that will give a unique prime output for every distinct prime input? Analyticity should preclude cheap reiterating upon a prime-counting "function", but permiss non-elementary mathematical operation. I’m curious what the a priori maximal and (non-trivial )minimal domain and especially codomain are of various $f()$ (i.e. without $$ predefined as a non-composite integer), and if there any $f$ with the full desired codomain that are not monotonic over its fully defined (ostensibly equivalent) domain —despite being injective, but presumably not wholly surjective.

If such mappings do exist, then those with the greatest simplicity and lowest computational complexity ought to be sought. Also minimizing the magnitude of the growth rate would be preferred (e.g. minimal $\mathrm{deg}(f(p))$ for algebraic polynomial $f$), while retaining self-definedness.

user1057382
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  • So, to rephrase your question, you are searching for function which map primes to primes and are in some sense easily definable without prime counting functions? I'm not sure what analytic is supposed to mean in this context. – Sven-Ole Behrend May 15 '22 at 17:03
  • How about the identity function (on $\mathbb C$, I presume)? – Lee Mosher May 15 '22 at 17:03
  • @Sven-Ole Behrad. Perhaps analyticity is not strictly what I be seeking, but.. representability? Given an input there is a well-defoned procedure to an single output. – user1057382 May 15 '22 at 17:06
  • @Lee Mosher Is said function injective? – user1057382 May 15 '22 at 17:06
  • Identity function – Lee Mosher May 15 '22 at 17:07
  • @Lee Mosher. How does that relate to $\mathbb{Z}$, let alone primes-subclass thereof? – user1057382 May 15 '22 at 17:09
  • Look up https://math.stackexchange.com/questions/2365192/analytic-lagrange-interpolation-for-a-countably-infinite-set-of-points . This would imply that e.g. there is a holomorphic (analytic on $\mathbb C$) function that maps every prime number $p_n$ into the next prime number $p_{n+1}$, i.e. $2\mapsto 3\mapsto 5\mapsto 7\mapsto 11\mapsto\ldots$. I don’t know if such function has been described/researched and what’s been found out about it, though! –  May 15 '22 at 17:16
  • And, trivially, the identity function maps prime numbers to prime numbers, that is, every prime number is mapped to itself. (You never said that was not allowed!) –  May 15 '22 at 17:20
  • @Stinking Bishop. You're right. Now I see the validity in your & Lee Mosher's proposal. Although, the identity function is sort of trivial, no? – user1057382 May 15 '22 at 17:24
  • @Stinking Bishop. Does this suggest that there does not exist an $f$ as described? or perhaps just not a continuous, if not smooth, one? – user1057382 May 15 '22 at 17:26
  • Someone suggested this question is answered at, or could be subsumed by, https://math.stackexchange.com/questions/2365192/analytic-lagrange-interpolation-for-a-countably-infinite-set-of-points. Could someone explain briefly how? I would be glad to approve the merge and closure if someone else does not first. – user1057382 May 15 '22 at 17:30
  • I suggested that - see my previous comments. –  May 15 '22 at 18:01
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    I agree that the identity function is a trivial answer. So if in retrospect you don't like that your question has such a trivial answer, this is an indication of the need to rethink the formulation of your question. – Lee Mosher May 15 '22 at 18:35
  • @Lee Mosher. Any ideas? Mewants to learn. – user1057382 May 15 '22 at 23:02

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