Find positive integers $(a,b,c)$ such that $n$ is a quadratic residue modulo some prime $p$ implies $an^2+bn+c$ is also a quadratic residue modulo $p$.
What I did:
I put $n=m^2$.Then we see $n$ is a quadratic residue modulo every prime $p$.So is $am^4+bm^2+c$ but the parts I am not sure about is:
1)if an integer is perfect square modulo every prime does it follow that it is a perfect square?
2)If a polynomial is perfect square at every integer then is it a perfect square in $\mathbb{Z}[x]$?
3)And suppose 1 and 2 are true.Then we get $(a,b,c)=(d^2,2de,e^2)$ for some $d,e\in \mathbb{N}$ but is that all solutions?
Now for the first two everyone I asked they said that they are true but could not produce a proof of any one of them.And for the third I am not sure but I believe that will be indeed all the solutions but I have no way to prove it.So any one post a solution please help me with clear and explained solution I mean no step jumps or claiming "this is trivial" or "well known" because apparently it is not to me.So I will ask you people to please give a detailed proof.Thanks in advance.
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shadow10
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Hagen Von Eitzen gave an answer on this question: http://math.stackexchange.com/questions/646094/is-every-non-square-integer-a-primitive-root-modulo-some-odd-prime. The question involving primitive roots remains unsolved, but the one you're looking for is "If an integer is a square modulo every prime, then is it a square itself?", which was (non-elementary, unfortunately) answered. This deals, of course, with only a small part of your question. – Bart Michels Feb 04 '14 at 12:09
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Thank you but no solution on the polynomial part.Can you help me please? – shadow10 Feb 04 '14 at 16:28