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Every root of $−1$ quadratic residue modulo $p$ prime, $p=1(\mod4)$ is distinct.

Running tests it appears that some values are never root of $−1$ quadratic residue modulo $p$.

For exemple : $7, 18, 21, 38, 41$ etc.

Is there a way to "predict" which number will never be a root ?

Best regards.

Arthur
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BenLaz
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  • Do you want numbers $x$ such that $0 < x < p$ ? If not, then $p=5$ works for $x=7$. – lhf Dec 20 '16 at 11:31
  • Your question looks like it was cut and pasted (poorly) from elsewhere. Please edit. Also, the question is unclear. for any $x$ there is some prime $p$ such that $x^2\equiv -1 \pmod p$ (just take any $p$ which divides $x^2+1$.). – lulu Dec 20 '16 at 11:38
  • Yes the numbers x are 0<x<p. – BenLaz Dec 20 '16 at 12:02
  • This question is not a copy / paste from elsewhere, it's a question i ask myself. – BenLaz Dec 20 '16 at 12:04

1 Answers1

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The question seems to be:

Which $n$ are not square roots of $-1$ mod $p$ for some prime $p > n$ ?

This is the same as

Which $n$ have the property that the largest prime factor of $n^2+1$ is less than $n$ ?

The sequence of such $n$ is listed at OEIS as A256011. Nothing much seems to be known about it.

lhf
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