Squaresum

Squaresum is the term used for a sum of squares of integers. In general, a squaresum expresses a natural number n as n = a1^2 + a2^2 + ... + ak^2, where the ai are integers and k is a positive integer. This concept lies at the heart of classical results in number theory and has geometric interpretation as the squared Euclidean length of a vector.

A central topic is representability as a sum of two squares. Fermat’s two-square theorem states that a positive integer n can be written as n = x^2 + y^2 if and only if every prime p congruent to 3 modulo 4 occurs in the prime factorization of n with even exponent. For example, 25 = 0^2 + 5^2 and 65 = 1^2 + 8^2, while 3 and 6 are not sums of two squares.

Beyond two squares, Legendre’s three-square theorem characterizes sums of three squares: a natural number n is a sum of three squares if and only if n is not of the form 4^a(8b+7). Lagrange’s four-square theorem goes further, asserting that every natural number is a sum of four squares.

The two-square case also has a representation-count formula: the number r2(n) of representations of n as a^2 + b^2 (counting signs and order) equals 4 times the difference between the number of divisors of n congruent to 1 mod 4 and those congruent to 3 mod 4. This connects to divisor theory and can be used to estimate representations.

Squaresums appear in geometry as the squared norm of vectors and in various algorithms for finding representations or testing feasibility. Examples include 50 = 1^2 + 7^2 and 85 = 2^2 + 9^2. See also sum of squares, Pythagorean triples, and norm.