<s>[INST] Find the remainder of $2^n+n^2$ modulus 6 
Find the remainder of $2^n+n^2$ modulus 6 given that $2^n+n^2$ is a prime and $n\geq2$($n$ positive integer)  

I tried to solve this but failed!I just know that $n$ must be odd. No progress at all!!
 [/INST] 2^n is never odd, so n^2 has to be odd to give a prime, n is odd
2^n with n odd is 4^k 2, 4^k = 4 mod 6, so 
2^n = 2 mod 6 here
n is 1, 3 or 5 mod 6
n^2 is 1 or 3 mod 6
if n^2 = 1 mod 6, then 2^n + n^2 = 3 mod 6, which would mean that when divided by 6, then 3 would remain, meaning that 2^n + n^2 divides by 3 and is not prime
so n^2 = 3 mod 6
2^n + n^2 = = 5 mod 6 
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