function for quadratic residue
- Fm Jakaria
- Posts:79
- Joined:Thu Feb 28, 2013 11:49 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.
You cannot say if I fail to recite-
the umpteenth digit of PI,
Whether I'll live - or
whether I may, drown in tub and die.
the umpteenth digit of PI,
Whether I'll live - or
whether I may, drown in tub and die.
Re: function for quadratic residue
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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Nur Muhammad Shafiullah | Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Re: function for quadratic residue
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$a$ 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.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$.
One one thing is neutral in the universe, that is $0$.