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Lagrange's four-square theorem
Lagrange's four-square theorem, also known as Bachet's conjecture, states that every nonnegative integer can be represented as a sum of four non-negative integer squares. That is, the squares form an additive basis of order four: where the four numbers are integers. For illustration, 3, 31, and 310 can be represented as the sum of four squares as follows:
![{\displaystyle {\begin{aligned}3&=1^{2}+1^{2}+1^{2}+0^{2}\\[3pt]31&=5^{2}+2^{2}+1^{2}+1^{2}\\[3pt]310&=17^{2}+4^{2}+2^{2}+1^{2}\\[3pt]&=16^{2}+7^{2}+2^{2}+1^{2}\\[3pt]&=15^{2}+9^{2}+2^{2}+0^{2}\\[3pt]&=12^{2}+11^{2}+6^{2}+3^{2}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55818f31dee3cb222e1be20630fee8b6aa25f3b4)
This theorem was proven by Joseph-Louis Lagrange in 1770. It is a special case of the Fermat polygonal number theorem.
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 extended 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 for integers k and 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.
The formula is also linked to Descartes' theorem of four "kissing circles", which involves the sum of the squares of the curvatures of four circles. This is also linked to Apollonian gaskets, which were more recently related to the Ramanujan–Petersson conjecture.
Several very similar modern versions of Lagrange's proof exist. The proof below is a slightly simplified version, in which the cases for which m is even or odd do not require separate arguments.
It is sufficient to prove the theorem for every odd prime number p. This immediately follows from Euler's four-square identity (and from the fact that the theorem is true for the numbers 1 and 2).
The residues of a2 modulo p are distinct for every a between 0 and (p − 1)/2 (inclusive). To see this, take some a and define c as a2 mod p. a is a root of the polynomial x2 − c over the field Z/pZ. So is p − a (which is different from a). In a field K, any polynomial of degree n has at most n distinct roots (Lagrange's theorem (number theory)), so there are no other a with this property, in particular not among 0 to (p − 1)/2.
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Lagrange's four-square theorem
Lagrange's four-square theorem, also known as Bachet's conjecture, states that every nonnegative integer can be represented as a sum of four non-negative integer squares. That is, the squares form an additive basis of order four: where the four numbers 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. It is a special case of the Fermat polygonal number theorem.
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 extended 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 for integers k and 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.
The formula is also linked to Descartes' theorem of four "kissing circles", which involves the sum of the squares of the curvatures of four circles. This is also linked to Apollonian gaskets, which were more recently related to the Ramanujan–Petersson conjecture.
Several very similar modern versions of Lagrange's proof exist. The proof below is a slightly simplified version, in which the cases for which m is even or odd do not require separate arguments.
It is sufficient to prove the theorem for every odd prime number p. This immediately follows from Euler's four-square identity (and from the fact that the theorem is true for the numbers 1 and 2).
The residues of a2 modulo p are distinct for every a between 0 and (p − 1)/2 (inclusive). To see this, take some a and define c as a2 mod p. a is a root of the polynomial x2 − c over the field Z/pZ. So is p − a (which is different from a). In a field K, any polynomial of degree n has at most n distinct roots (Lagrange's theorem (number theory)), so there are no other a with this property, in particular not among 0 to (p − 1)/2.