Paper 4, Section II, D

Numbers and Sets | Part IA, 2017

(a) State and prove the Fermat-Euler Theorem. Deduce Fermat's Little Theorem. State Wilson's Theorem.

(b) Let pp be an odd prime. Prove that X21(modp)X^{2} \equiv-1(\bmod p) is solvable if and only if p1(mod4)p \equiv 1(\bmod 4).

(c) Let pp be prime. If hh and kk are non-negative integers with h+k=p1h+k=p-1, prove that h!k!+(1)h0(modp).h ! k !+(-1)^{h} \equiv 0(\bmod p) .

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