<s>[INST] Only finitely many $a, b$ such that $2+3^n+5^{n^2}=2^a7^b$ for some $n$? Let $(a,b)$ be a pair of positive integers such that $$2+3^n+5^{n^2}=2^a7^b$$ for some positive integer $n$. Is it true that there are only finitely many such pairs?
I don't know the answer to such question, but I guess that it is positive..
 [/INST] No positive integer solutions exist to $2+3^n+5^{n^2}=2^a7^b$.
$n$ even $\,\Rightarrow\, 2+3^n+5^{n^2}\equiv 2+(-1)^{n^2}\equiv 2+1\equiv 0\not\equiv 2^a7^b\pmod{\!3}$.   
$n$ odd $\,\Rightarrow\, 2+3^n+5^{n^2}\equiv 2+(-1)^n+1^{n^2}\equiv 2\equiv 2^a7^b\pmod{\!4}$, so $a=1$.    
But $2+3^n+5^{n^2}\equiv 2+(-1)^{n^2}\equiv 1\not\equiv 2\cdot 7^b\equiv 2\cdot 1^b\equiv 2\pmod{\!3}$.
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