Lagrange's four-square theorem

Every nonnegative integer is a sum of four integer squares. The theorem is sharp: some integers, such as 7, cannot be represented as sums of three squares.

Four-square identity: The product of two sums of four squares is itself a sum of four squares. It lets a proof reduce the theorem to representing prime numbers as four squares.
Perfect square: An integer equal to the square of another integer. Each summand in the theorem is a perfect square, including zero.
Joseph-Louis Lagrange: An Italian-born mathematician whose work shaped number theory, mechanics, and mathematical analysis. He proved the four-square theorem in 1770.
Waring's problem: The problem of finding, for each positive integer k, a least number of kth powers sufficient to represent every nonnegative integer. The four-square theorem gives the exact answer for squares: four suffice.
Seven as a sum of four squares: The representation 7 = 2² + 1² + 1² + 1². Seven is the smallest positive integer that cannot be expressed using three squares.
Hurwitz quaternion: A quaternion with integer or half-integer coordinates belonging to the Hurwitz order. Quaternion multiplication supplies the algebraic form of the four-square identity.
Integer partition: A way of writing a positive integer as a sum of positive integers, disregarding order. A four-square representation is a constrained additive decomposition with square summands.
Leonhard Euler: An eighteenth-century Swiss mathematician whose work spans number theory, analysis, and mechanics. Euler discovered the four-square identity and developed related arguments before Lagrange's proof.
Lagrange's four-square theorem algorithm: An algorithm that finds four integer squares summing to a given nonnegative integer. Constructive proofs can be turned into procedures for finding representations.
Zero as a sum of four squares: The representation of zero as 0² + 0² + 0² + 0². Zero is included in the theorem through the allowance of zero coordinates.

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