Theorem

Legendre's Three-Square Theorem

Number Theory

Legendre's Three-Square Theorem states that a natural number can be written as a sum of three integer squares if and only if it is not of the form four to a power times an integer of the form eight times a whole number plus seven. Named for Adrien-Marie Legendre, who first stated it, it identifies exactly which whole numbers require a fourth square beyond three, complementing Lagrange's Four-Square Theorem, and the full proof of the 'if' direction was later completed rigorously by Carl Friedrich Gauss.

Facts

Classification

Statement Form

Existence Theorem 1

Statement Form

Characterization Theorem 1

Partially Attested

Proof Year

1797 1

Source gives 1797 or 1798 for Legendre's first proof.

Statement

A natural number can be written as a sum of three squares of integers if and only if it is not of the form 4^a(8b+7). 1

Connections

Has Statement Form

Characterization

Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.

Existence Theorem

Named After

Adrien-Marie Legendre

Derived from the theorem's own name (unambiguous possessive-token match to exactly one live mathematician entity, w-bfill-g5-0924 browse backfill)

Proved By

Adrien-Marie Legendre

Sources

1. Legendre's three-square theorem (Wikipedia)

Statement section
a natural number can be represented as the sum of three squares of integers n = x² + y² + z² if and only if n is not of the form n = 4^a(8b + 7)
History section
In 1797 or 1798 A.-M. Legendre obtained the first proof of his 3 square theorem.

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Legendre's Three-Square Theorem

Has Statement Form

Named After

Proved By