neliösummana

Neliösummana, in Finnish mathematical usage, refers to expressing an integer as a sum of square numbers. The concept is a central topic in additive number theory, with attention to representations by two, three, or four squares and to related multiplicative properties.

Two-squares, three-squares, and four-squares theorems provide key characterizations. Fermat’s two-square theorem states that a prime p can be written as p = a^2 + b^2 if and only if p ≡ 1 (mod 4). More generally, a number is a sum of two squares precisely when, in its prime factorization, every prime congruent to 3 mod 4 occurs with an even exponent. Legendre’s three-square theorem gives a complete criterion for three-square representations: a positive integer 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 asserts that every natural number can be represented as a sum of four squares: n = a^2 + b^2 + c^2 + d^2 for some integers a, b, c, d.

Several identities underpin the theory of neliösummana. Euler’s four-square identity shows that the product of two sums of four squares is itself a sum of four squares, reflecting a multiplicative structure. The problem also connects to algorithmic approaches for finding explicit representations and to broader topics in number theory, such as Gaussian integers for two-square representations.

Example representations illustrate the idea: 29 = 5^2 + 2^2 + 1^2 + 1^2, and 7 = 2^2 + 1^2 + 1^2 + 1^2 is a sum of four squares but not of three. Applications span mathematics, computer science, and cryptography.