Sum of two squares theorem

In number theory, the sum of two squares theorem relates the prime decomposition of any integer n > 1 to whether it can be written as a sum of two squares, such that n = a² + b² for some integers a, b.

An integer greater than one can be written as a sum of two squares if and only if its prime decomposition contains no prime congruent to ${\displaystyle 3}$ modulo ${\displaystyle 4}$ raised to an odd power.[1]

This theorem supplements Fermat's theorem on sums of two squares 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.

Examples

The prime decomposition of the number 2450 is given by 2450 = 2 · 5² · 7². Of the primes occurring in this decomposition, 2, 5, and 7, only 7 is congruent to 3 modulo 4. Its exponent in the decomposition, 2, is even. Therefore, the theorem states that it is expressible as the sum of two squares. Indeed, 2450 = 7² + 49².

The prime decomposition of the number 3430 is 2 · 5 · 7³. This time, the exponent of 7 in the decomposition is 3, an odd number. So 3430 cannot be written as the sum of two squares.

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. Dudley, Underwood (1969). "Sums of Two Squares". Elementary Number Theory. W.H. Freeman and Company. pp. 135–139.