<s>[INST] Let $x$ be an integer and $n$ be a positive integer. Find the smallest $n$ such that $x^4+n^2$ is not a prime for any $x$. I need help proving the following: Let $x$ be an integer and $n$ be a positive integer. Find the smallest $n$ such that $x^4+n^2$ is not a prime for any $x$. I know that the smallest $n$ is 8 by testing $n=1,2,3,...$. I need help proving that $n=8$ is the smallest $n$ such that $x^4+n^2$ is not a prime for any $x$. 
 [/INST] It is well-known that $x^4+4=\Big(x^2\Big)^2+2^2+\underbrace{2\cdot2x^2-2\cdot2x^2}_0=\Big(x^2+2\Big)^2-4x^2=$
$=\big(x^2-2x+2\big)\big(x^2+2x+2\big)$. However, since $x^2\pm2x+2=\pm1$ has as viable integer 
solutions $x=\pm1$, we must extend the search to polynomials of the form $x^2+4a^4$, which 
has no such solutions for $a=2\iff n=8$.
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