 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2011/c/c2/a/ii/solution

p

and

p ∤ a

a^{(p - 1) /2} \equiv (\frac{a}{p}) (mod p) .

Thus this power is congruent to one for a nonzero square and to minus one for a nonsquare. The congruence also holds when

p ∣ a

if both sides are taken to be zero.

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