thumb|Unlike in three dimensions in which distances between [[Vertex (geometry)|vertices of a polycube with unit edges excludes √7 due to Legendre's three-square theorem, Lagrange's four-square theorem states that the analogue in four dimensions yields square roots of every natural number ]]

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:

<math display="block">p = a^2 + b^2 + c^2 + d^2,</math>

where the four numbers <math>a, b, c, d</math> are integers. For illustration, 3, 31, and 310 can be represented as the sum of four squares as follows:

<math display="block">\begin{align}

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{align}</math>

This theorem was proven by Joseph-Louis Lagrange in 1770. It is a special case of the Fermat polygonal number theorem.

Historical development

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 <math>4^k(8m+7)</math> for integers and . 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.

Proofs

The classical proof

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.

Proof using the Hurwitz integers

Another way to prove the theorem relies on Hurwitz quaternions, which are the analog of integers for quaternions.

m&\text{if }n\text{ is even}.

\end{cases}</math>

Equivalently, it is eight times the sum of all its divisors which are not divisible by 4, i.e.

<math display="block">r_4(n)=8\sum_{m\,:\, 4\nmid m\mid n}m.</math>

We may also write this as

<math display="block">r_4(n) = 8 \sigma(n) -32 \sigma(n/4) \ , </math>

where the second term is to be taken as zero if n is not divisible by 4. In particular, for a prime number p we have the explicit formula&nbsp;.

Some values of r<sub>4</sub>(n) occur infinitely often as whenever n is even. The values of r<sub>4</sub>(n)/n can be arbitrarily large: indeed, r<sub>4</sub>(n)/n is infinitely often larger than 8. proved that each natural number can be written as a sum of four squares with some requirements on the choice of these four numbers.

One may also wonder whether it is necessary to use the entire set of square integers to write each natural as the sum of four squares. Eduard Wirsing proved that there exists a set of squares with <math>|S| = O(n^{1/4}\log^{1/4} n)</math> such that every positive integer smaller than or equal to can be written as a sum of at most 4 elements of .

See also

  • Fermat's theorem on sums of two squares
  • Fermat's polygonal number theorem
  • Waring's problem
  • Legendre's three-square theorem
  • Sum of two squares theorem
  • Sum of squares function
  • 15 and 290 theorems

Notes

</references>

References

  • Proof at PlanetMath.org
  • Another proof
  • An applet decomposing numbers as sums of four squares
  • OEIS index to sequences related to sums of squares and sums of cubes