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 term p^k, where prime ${\displaystyle p\equiv 3{\pmod {4}}}$ and k is odd.[1]

If ${\displaystyle n}$ is a perfect square, it can be written with the trivial case applied, setting ${\displaystyle a}$ (or ${\displaystyle b}$) to zero: ${\displaystyle n^{2}=0^{2}+n^{2}}$ (Some perfect squares also have non-trivial cases such as ${\displaystyle 25=4^{2}+3^{2}}$ and ${\displaystyle 100=8^{2}+6^{2}}$, also known as Pythagorean triples).

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.
• Lagrange's four-square theorem
• Landau–Ramanujan constant, used in a formula for the density of the numbers that are sums of two squares
• Legendre's three-square theorem

References

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