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I thought something fancy can be done with number-theory or memoization, but neither worked for me. Being limited in knowledge I decided to ask experts.

Does there exist a deterministic polynomial-space algorithm for the following decision problem?

Instance: Two positive integers $k$ and $m$ in decimal representation.
Question: Is $2^k+m$ a prime number?

Some comments:

  • By polynomial-space I mean that the space complexity of the algorithm should be bounded by a polynomial in the input length $\log k+\log m$.

  • The naive approach to this problem determines the decimal representation of $2^k+m$ and then applies a fast primality testing algorithm. It is easy to see that this naive approach requires exponential space, just for writing down the decimal representation of $2^k+m$.

  • This is related to another question of mine.

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    $\begingroup$ (i) By "polynomial space" do you mean polynomial in the size of the (presumably binary) encoding of $K$ and $M$? That is, polynomial in $\log K + \log M$? (ii) "Calculating $k\mapsto 2^k$ takes exponential time" is a strange thing to say. Do you just mean representing $2^k$ takes space exponential in $\log k$? That's sort of obvious, I think, so doesn't help clarify your post or why it is interesting (my opinion). And the post you link to, and its answers, reflect a similar confusion. But I think the question you ask is interesting. Editing the post to clarify it would help. $\endgroup$
    – Neal Young
    Sep 2, 2020 at 1:10
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    $\begingroup$ Anyway, as far as I am aware, no such algorithm is known even in the special case of Fermat primes ($M=1$, $K$ a power of $2$). $\endgroup$ Sep 2, 2020 at 6:17
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    $\begingroup$ All problems in $NP$ are in $PSPACE$, just by showing its in $NP$ is enough to prove a deterministic poly-space algorithm exists. Whether one is known is apparently open. $\endgroup$
    – The T
    Sep 2, 2020 at 15:42
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    $\begingroup$ A positive answer here would in particular falsify the more general “no formula for primes” conjecture here: cstheory.stackexchange.com/questions/46902/…, and least given a modest extra assumption. $\endgroup$ Sep 3, 2020 at 14:47
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    $\begingroup$ Of course you can it in polynomial time in the number of bits of the prime, but this is too slow to resolve either of our questions. $\endgroup$ Sep 4, 2020 at 7:32

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