Are there infinitely many primes $p$, for which there exist positive integers $m$ and $n$, such that $2^m+n^2=p$?
2026-04-29 20:23:18.1777494198
Infinitely many primes that are the sum of a power of 2 and a perfect square
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Not an answer, but my link didn't work in a comment :(
Have you considered the easier question of whether or not there are infinitely many primes of the form $2+n^2$? Note that $$2+1^2=3, 2+3^2=11, 2+9^2=83,2+15^2=227, 21^2+2=443,33^2+2=1091,39^2+2=1523,45^2+2=2027,57^2+2=3251,81^2+2=6563$$ are primes, I can imagine there are infinitely many of those already.
Typically these are open problems. I refer you to this question and it's answer for more on that.