# Prove that if p is a prime number of the form 4n + 1

Prove that if p is a prime number of the form 4n + 1,

then the following congruence holds:

x^2 is congruent to -1 mod p.

The more detail problem is in the attached file.

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#### Solution Preview

The solution is in the attached file.

Ring Theory 3

Solving Congruences

By :- Thokchom Sarojkumar Sinha

Prove that if p is a prime number of the form 4n + 1

then the following congruence holds,

Proof :- Suppose that

Since p = 4n + 1 then p - 1 = 4n

which implies that

Therefore the product for ...

#### Solution Summary

The solution explains the congruence: x^2 is congruent to -1 mod p.

The solution is given in detail.

This is mainly for solving the congruence by using the Wilson's theorem.