<s>[INST] If $n$ is a positive integer such that $2^n+n^2$ is a prime number , then is it true that $6|n-3$ ? If $n$ is a positive integer such that $2^n+n^2$ is a prime number , then is it true that $6|n-3$ ? Trivially $n$ cannot be even , so this leaves us only with the possibilities $n \equiv1,3,5( \mod 6)  $ , but I cannot reduce the cases . Please help , Thanks in advance . 
 [/INST] If $n\equiv1,5\pmod 6$ then $n^2\equiv 1\pmod 6$.
On the other hand, $2^n\equiv 2$ or $4\pmod 6$ depending on $n$ is odd or even. So if $n^2\equiv 1\pmod 6$, then
$$2^n+n^2\equiv 2+1\equiv 3\pmod 6$$
therefore $2^n+n^2$ is a multiple of three.
Notice the case $1^2+2^1$, that is the only exception to your statement.
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