Lagrange's four-square theorem

The four- square theorem or Lagrange's theorem is a sentence from the mathematical branch number theory. The sentence reads:

A few examples:

There are numbers for which there are multiple representations as a sum of four square numbers. Thus, the 310 can, for example, be expressed as follows:

The statement of the theorem of Lagrange was already suspected in 1621 Bachet and 1640 by Pierre de Fermat. Joseph Louis Lagrange published in 1770 the first evidence. This was three years later simplified considerably by Leonhard Euler.

Natural numbers as sums of squares

There are natural numbers that can be represented as a sum of two squares: So, for example, 20 = 16 4 for 21, however, there is not such a representation.

In general, a natural number n then not as a sum of two square numbers is displayed when the prime factorization of n contains at least one prime p in odd multiplicity, in which: p is congruent to 3 modulo 4, or in mathematical notation.

Example: 14 = 2 * 7 7 is regarding 4 in the residue class 3 So there can be no representation of 14 give the sum of two square numbers. By contrast, 98 = 2 * 7 * 7 Although applies here also that 7 is regarding 4 in the residue class 3, but duplicated in the prime factorization, so there may be a representation of 98 as a sum of two squares, namely 49 49.

Conversely Fermat has been found that every prime p, in which: when the sum of two squared numbers can be displayed. This knowledge was used by the mathematician Carl Gustav Jacob Jacobi, to prove the theorem:

Any natural number n is exactly then represented as the sum of two squares if all occur in a straight multiplicity in the prime factorization of n.

The German mathematician Edmund Landau showed that the number of such numbers that can be represented as the sum of two square numbers is relatively small.

What is interesting is the question of how many summands at most are needed to represent any natural number as a sum of squares. This question is answered the above- illustrated four- square theorem.

Related topic: Euler four- square theorem

If one has with

The representations of two numbers n1 and n2 as the sum of four squares, then one has over the quaternions

Also a representation of the product

As the sum of four squares. This identity had already discovered Leonhard Euler in 1748, is named after him also Euler four- square theorem. With this sentence he reduced the proof of the theorem that every number can be written as the sum of four square numbers, prime number. Are namely prime numbers representable as a sum of four squares, as well as products of prime numbers; so also all natural numbers, since they are products of primes.

Extension of the problem

The sentence was extended in 1798 by Adrien -Marie Legendre to the three -squares set by he found out that every natural number a maximum of three square numbers can be composed, in the event that they can not be represented in the form. A gap in the proof was later closed by Carl Friedrich Gauss, which is why he is also known as Gauss. Peter Gustav Lejeune Dirichlet and Edmund Landau found simplifications of the proof.

As an advanced question is now formulated the Waring problem, how you can specify the minimum number of summands necessary to represent each natural number as the sum of numbers with exponents.

Number of views

As already shown in the introduction, there are sometimes multiple representations of a number as a sum of four square numbers. A formula for the number of such representations provides the set of Jacobi.