Alternative proof for Fermat’s Two-Square Theorem

Proof.

Assume \( p \) is an integer prime that is congruent to \( 1\bmod 4 \). Then \( \exists n,r \in \mathbb Z : n^2+1=pr \implies p|(n+i)(n-i) \) Note \( p \) does not divide either \( n+i \) or \( n-i \). So \( p \) is not a prime in \( \mathbb Z [i] \). Then there exists a prime \( q \in \mathbb Z [i] : q|p \).

\( \implies N(q)|N(p)=p^2 \) since \( p,q \) are not associates \( \implies N(q)=N(q_1+q_2 i)=q_1^2+q_2^2=p \) where \( q_1, q_2 \) are integers

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Note we can find \( q=\gcd(p,n+i) \) by Euclid’s algorithm.