Numbers and Sets | Part IA, 2001

(a) Prove that, if pp is prime and aa is not a multiple of pp, then ap11(modp)a^{p-1} \equiv 1(\bmod p).

(b) The order of a(modp)a(\bmod p) is the least positive integer dd such that ad1(modp)a^{d} \equiv 1(\bmod p). Suppose now that ax1(modp)a^{x} \equiv 1(\bmod p); what can you say about xx in terms of dd ? Show that p1(modd)p \equiv 1(\bmod d).

(c) Suppose that pp is an odd prime. What is the order of x(modp)x(\bmod p) if x21(modp)x^{2} \equiv-1(\bmod p) ? Find a condition on p(mod4)p(\bmod 4) that is equivalent to the existence of an integer xx with x21(modp)x^{2} \equiv-1(\bmod p).

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