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I'm slowly reading through the book What is Mathematics which asks the following question at the end of its quadratic residues section. I'm not sure how to begin it really, so any hints/suggestions would be greatly appreciated.

1. The problem statement, all variables and given/known data

We have seen that [tex] x^{2} \equiv (p - x)^{2} \pmod p [/tex]. Where p is a prime > 2 and x is not divisible by p

Show that these are the only congruences among the numbers [tex]1^{2}, 2^{2}, 3^{2},...,(p-1)^{2}[/tex]

2. Relevant equations

[tex] (p-x)^{2} = p^{2} - 2px + x^{2}\equiv x^{2} \pmod p [/tex]

3. The attempt at a solution

No idea..

thanks in advance,

babak

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# Homework Help: Congruences of quadratic residues

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