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I would like to prove the following claim which I think is true:

Claim: Let $n,$ $m$ and $s$ be positive numbers. Fix $s$, then for every positive number $n$ the $\gcd(1+n+n^2,1+n+s+n^2+2ns+s^2)$ will be equal to a divisor of $1+5s^2+s^4.$

For example for every positive number $n$ if we set $s=8$ the $\gcd(1+n+n^2,73+n+16n+n^2)=1,7,631$ or $4417$. We can see that $1+5*8^2+8^4=4417$ and $4417=7*631$.

I came to the claim by moving numbers around in GAP. If it is wrong a counter example would be awesome.

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Take $n=1$ and $s=3$. Then $$\displaylines{ 1+n+n^2=3\cr 1+n+n^2+s+2ns+s^2=21\cr \gcd(3,21)=3\cr 1+5s^2+s^4=127\cr 3\ \hbox{is not a factor of}\ 127\ .\cr}$$

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    You have the correct answer ! I just needed a counterexample. – 2017-01-18
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No way a statement like this can be right.

$\gcd(1+n+n^2, 1+n+s+n^2+2ns+s^2) = \gcd(1+n+n^2, s(1+2n+n^2)) = \gcd(1+n+n^2, sn) = \gcd(1+n+n^2, s)$. So simply take $s = n^2+n+1$, then it can never be $s |1+5s^2+s^4$, unless $s=1$.

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    Perhaps you can show a specific example? – 2017-01-18
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    I have shown you an infinite family of counter examples - that is not good enough? – 2017-01-18
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    @David It is strange. I have misclicked on the mobile app and strangely can't retract the vote! – 2017-01-18
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    @S.C.B. According to my browser the downvote has disappeared, so I think it's OK. – 2017-01-18