Legendre's three-square theorem

A positive integer is a sum of three integer squares exactly when it is not of the form 4ᵃ(8b + 7), for nonnegative integers a and b.

Three-square theorem: A theorem characterizing which nonnegative integers can be expressed as sums of three integer squares. This broader formulation places Legendre’s result within the general problem of representing integers by three squares.
Sum of squares: An expression formed by adding squares of integers or other specified quantities. The theorem is a precise existence criterion for one particular number of integer squares.
Adrien-Marie Legendre: A French mathematician whose work included number theory, analysis, and geometry. Legendre gave an early proof of the three-square criterion bearing his name.
Legendre symbol: A function encoding whether an integer is a quadratic residue modulo an odd prime. Legendre symbols help analyze representation questions through congruences and quadratic forms.
Quadratic residues: Residue classes that are squares modulo a given integer. Squares modulo 8 restrict sums of three squares and expose the obstruction at numbers congruent to 7.
Perfect square: An integer equal to the square of another integer. Each term in a three-square representation is a perfect square, including zero.
Carl Friedrich Gauss: A German mathematician whose work established foundational results across number theory and other fields. Gauss later proved the three-square theorem in his systematic study of quadratic forms.
Theta function: A function defined by a series whose terms encode integer lattice points and quadratic-form representations. Theta series package counts of three-square representations into analytic objects.
Infinite descent: A proof method that derives a contradiction by constructing an indefinitely decreasing sequence of positive integers. Descent arguments help establish that integers outside the excluded family admit a representation.
Integer: A whole number that is positive, negative, or zero. The theorem concerns positive integers and allows the squared terms to be any integers.

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