 Solution

ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2023/1/b/ii/solution

For odd

p \neq = 5

, quadratic reciprocity has no sign factor because

5 \equiv 1 (mod 4)

(\frac{5}{p}) = (\frac{p}{5}) .

5

are

1, 4

, so the answer for odd primes is

p \equiv 1, 4 (mod 5)

. The excluded-prime wording also allows

p = 2

, where

5 \equiv 1

has a square root. Thus the full answer is

p = 2 or p \equiv 1, 4 (mod 5) .

 New to topics? Read the docs here!