Maths Olympiad Prep

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Number theory Difficulty 6.1 National olympiad Prove it

13. Show that if aa is a quadratic residue of the prime pp, then the solutions of x2a(modp)x^{2} \equiv a(\bmod p) are
a) x±an+1(modp)x \equiv \pm a^{n+1}(\bmod p), if p=4n+3p=4 n+3.
b) x±22n+1an+1(modp)\quad x \equiv \pm 2^{2 n+1} a^{n+1}(\bmod p), if p=8n+5p=8 n+5

Solution

None

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