# Number Theory Perfect Square

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Is there a perfect square n^2 such that

n^2 = -1 (mod p) for

p=3

p=5

p=7

p=11

p=13

p=17

p=19?

Can you characterize the primes for which n^2 = -1 (mod p) has a solution?

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#### Solution Preview

Sol: If a = b (mod k), then b = a + mk, where m is an integer.

Here, n^2 = -1 (mod p) implies that -1 = n^2 + mp, where m is an ...

#### Solution Summary

The number theory for perfect squares are determined.

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