Sum of two squares theorem

In number theory, the sum of two squares theorem says when an integer $n>1$ can be written as a sum of two squares, that is when $n=a^{2}+b^{2}$ for some integers $a,b$.

An integer greater than one can be written as a sum of two squares if and only if in its prime decomposition there is no prime congruent to 3 (mod 4) raised to an odd power.^[1]

This theorem supplements Fermat's two-square theorem which says when a prime number can be written as a sum of two squares, in that it also covers the case for composite numbers.

See also

• Brahmagupta–Fibonacci identity. This identity entails that the set of all sums of two squares is closed under multiplication.
• Landau–Ramanujan constant, used in a formula for the density of the numbers that are sums of two squares

References

1. Underwood Dudley (1978). Elementary Number Theory (2 ed.). W.H. Freeman and Company.