function for quadratic residue

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Fm Jakaria
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function for quadratic residue

Unread post by Fm Jakaria » Sun Nov 23, 2014 4:37 pm

Determine all integers(ALL) $a $ such that there exists an integral valued function $f $ depending on $a$, defined for all sufficiently large primes $ p$ (domain depends on $a$) ; $f(p)^2-a$ is divisible by $ p$.
Last edited by Fm Jakaria on Mon Nov 24, 2014 9:07 pm, edited 1 time in total.
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*Mahi*
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Re: function for quadratic residue

Unread post by *Mahi* » Mon Nov 24, 2014 8:51 pm

This question is meaningless without fixing a proper range for $f$ - for example, every $a$ satisfies the conditions if $f(x)=\sqrt {x+a}$.
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Masum
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Re: function for quadratic residue

Unread post by Masum » Wed Dec 17, 2014 8:16 pm

\
Fm Jakaria wrote:Determine all integers(ALL) $a $ such that there exists an integral valued function $f $ depending on $a$, defined for all sufficiently large primes $ p$ (domain depends on $a$) ; $f(p)^2-a$ is divisible by $ p$.
$a$ is a square itself. If $a$ is not a square, then $a=x^2\prod\limits_{p|a}p$ with $x^2$ the maximum square dividing $a$. Then use Jacobi symbol and Chinese Remainder Theorem, choose some prime wisely to show a contradiction.
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