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 Statement FormCharacterization Theorem 1 Partially Attested
Proof YearSource gives 1797 or 1798 for Legendre's first proof. StatementA 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
In Branch
Source Legendre's three-square theorem (Wikipedia)
Named After
Proved By
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.
In Branch: Number Theory, Lead
Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers
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