Theorem of the four squares of Lagrange

The theorem of the four squares of Lagrange , so known under the name of conjecture of Bachet and shown in 1770 by the Italian Mathématicien Joseph Louis Lagrange corresponds to a equation diophantienne which is solved with the techniques of the modular Arithmétique. It is stated in the following way:

All whole positive is expressed like the nap of to the more four square.

More formally, for entire positive N , there exist positive entireties has , B , C , D such as:

$n\; =\; a^2\; +\; b^2\; +\; c^2\; +\; d^2\; \backslash ,\; \backslash !$

The French mathematician Adrien-Marie Legendre improved the theorem in 1798 by affirming that a positive entirety can be expressed as the sum of with more the three squares if and only it is not form

$4^\; \{K\}\; (8m\; +\; 7)\; \backslash ,\; \backslash !$.

Its demonstration was incomplete, leaving a breach which was filled later by the German Mathématicien Carl Friedrich Gauss.

The theorem of the four squares of Lagrange is a particular case of the Théorème of the polygonal number of Fermat and Problème of Waring.

The demonstration of the theorem rests (partly) on the Identité of the four squares of Euler:

$(x\_1^2\; +\; y\_1^2\; +\; z\_1^2\; +\; t\_1^2)\; (x\_2^2\; +\; y\_2^2\; +\; z\_2^2\; +\; t\_2^2)\; =\; (x\_1x\_2\; +\; y\_1y\_2\; +\; z\_1z\_2\; +\; t\_1t\_2)\; ^2\; +\; (x\_1y\_2\; -\; y\_1x\_2\; +\; t\_1z\_2\; -\; z\_1t\_2)\; ^2$

$+\; \backslash \; left.\; (x\_1z\_2\; -\; z\_1x\_2\; +\; y\_1t\_2\; -\; t\_1y\_2)\; ^2\; +\; (x\_1t\_2\; -\; t\_1x\_2\; +\; z\_1y\_2\; -\; y\_1z\_2)\; ^2\; \backslash \; right.$

Arithmetic functions

The arithmetic functions make it possible to obtain more general results. If one poses $r\_4\; (N)$ as being the number of way of breaking up $n$ in the form of a sum of 4 squares, one obtains the following result:

$\backslash \; sum\_\; \{n=0\}\; ^\; \{\backslash \; infty\}\; \{r\_4\; (N)\; x^n\}\; =\; \backslash \; left\; (\backslash \; sum\_\; \{n=0\}\; ^\; \{\backslash \; infty\}\; \{x^\; \{n^2\}\}\; \backslash \; right)\; ^4$, for .

With the help of the use of the series of Lambert, one from of deduced the theorem following, known as theorem of Jacobi:

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*,\; \backslash \; r\_4\; (N)\; =\; 8\; \backslash \; sum\_\; \{D|N,\; D\; \backslash \; not\; \backslash \; equiv\; 0\}\; \{D\}$

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