Lagrange's four-square theorem

Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number can be represented as the sum of four integer squares.

where the four numbers ${\displaystyle a_{0},a_{1},a_{2},a_{3}}$ are integers. For illustration, 3, 31 and 310 can be represented as the sum of four squares as follows:

This theorem was proven by Joseph Louis Lagrange in 1770.

From examples given in the Arithmetica it is clear that Diophantus was aware of the theorem. This book was translated in 1621 into Latin by Bachet (Claude Gaspard Bachet de Méziriac), who stated the theorem in the notes of his translation. But the theorem was not proved until 1770 by Lagrange.

Adrien-Marie Legendre completed the theorem in 1797–8 with his three-square theorem, by proving that a positive integer can be expressed as the sum of three squares if and only if it is not of the form ${\displaystyle 4^{k}(8m+7)}$ for integers ${\displaystyle k}$ and ${\displaystyle m}$. Later, in 1834, Carl Gustav Jakob Jacobi discovered a simple formula for the number of representations of an integer as the sum of four squares with his own four-square theorem.

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