Lagrange's four-square theorem

Lagrange's four-square theorem

(http://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem)

Lagrange's four-square theorem, also known as Bachet's conjecture, was proven in 1770 by Joseph Louis Lagrange. An earlier proof by Fermat was never published.

The theorem appears in the Arithmetica of Diophantus, translated into Latin by Bachet in 1621. It states that every positive integer can be expressed as the sum of four squares of integers. For example,

3 = 1 ² + 1 ² + 1 ² + 0 ²

31 = 5 ² + 2 ² + 1 ² + 1 ²

310 = 17 ² + 4 ² + 2 ² + 1 ².

More formally, for every positive integer n there exist integers x₁, x₂, x₃, x₄ such that

n = x ₁ ² + x ₂ ² + x ₃ ² + x ₄ ².

Adrien-Marie Legendre improved on the theorem in 1798 by stating that a positive integer can be expressed as the sum of three squares if and only if it is not of the form 4^k(8m + 7). His proof was incomplete, leaving a gap which was later filled by Carl Friedrich Gauss.

Lagrange's four-square theorem is a special case of the Fermat polygonal number theorem and Waring's problem.

Euler's four-square identity

(http://en.wikipedia.org/wiki/Euler%27s_four-square_identity)

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In mathematics, Euler's four-square identity says that the product of two numbers, each of which being a sum of four squares, is itself a sum of four squares. Specifically:

Euler wrote about this identity in a letter dated May 4, 1748 to Goldbach^[1][2] (but note that he used a different sign convention from the above). It can be proven with elementary algebra and holds in every commutative ring. If the a_k and b_k are real numbers, a more elegant proof is available: the identity expresses the fact that the absolute value of the product of two quaternions is equal to the product of their absolute values, in the same way that the Brahmagupta-Fibonacci two-square identity does for complex numbers.

The sign convention used above corresponds to the signs obtained by multiplying two quaternions. Other sign conventions can be obtained by changing any a_k to − a_k, b_k to − b_k, or by changing the signs inside any of the squared terms on the right hand side. For example, changing a₁ to − a₁, b₁ to − b₁, and changing the signs of the second, third, and fourth terms on the right hand side yields the alternate form:

The identity was used by Lagrange to prove his four square theorem. More specifically, it allows the theorem to be proven only for prime numbers.