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"Find all the positive integers $n$ satisfying $2^n+n^2 +25 = p^3$ where $p$ is a prime number." I proved that $n$ must be divisible by 6 and predicted $n=6, p=5$. However, I can't wrap it up. Thank you for all.

Noun
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    Not even seeing any other prime powers on that list, but I didn't search all that far. I note that, with $n=24$, the left hand factors as $3079\times 348731$ which suggests that no simple congruence arguments will get you further. Perhaps there is an algebraic factoring of the left hand? – lulu May 26 '23 at 10:54
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    The only non-negative integer $n\le 10^5$ , for which $2^n+n^2+25$ is a perfect power , if $n=6$ – Peter May 26 '23 at 10:58
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    Found an answer on AoPS thanks to Approach0: https://artofproblemsolving.com/community/c1068820h2165310p16080737 – Bruno B May 26 '23 at 11:22
  • Wow thank you so much – Noun May 26 '23 at 11:36

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