Is there any method that I can follow to calculate $x$ and $y$ such that $p= x^{2}+y^{2} $ where $p$ is a prime and $p \equiv 1 ({mod}\ 4)$. Fermats theorem of sum of two squares.
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Use the Euclidean algorithm in $,\Bbb Z[i],$ and Cornacchia, see this answer. – Bill Dubuque Mar 16 '14 at 14:18