Four squares set

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

Any natural number can be written as the sum of four square numbers.

Examples:

4 = 1 + 1 + 1 + 1 = 4 + 0 + 0 + 0

7 = 4 + 1 + 1 + 1

31 = 25 + 4 + 1 + 1 = 9 + 9 + 9 + 4

This statement was suspected by Bachet in his influential Diophant edition in 1621 and proved by Lagrange in 1770 , using an identity found by Euler in 1748 , which reduced the problem to prime numbers.

Natural numbers as the sum of square numbers

There are natural numbers that can be represented as the sum of two square numbers: B. 20 = 16 + 4. For 21, however, there is no such representation.

Since the square of an odd number is always , spoken congruently 1 modulo 4 or leaves the remainder 1 when dividing by 4, the general rule is that a natural number cannot be represented as the sum of two square numbers if the prime factorization of at least one prime number in an odd multiple contains, for which applies: ${\displaystyle \equiv 1\mod 4}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle p}$

${\displaystyle p\equiv 3\mod 4}$.

Examples:

14 = 2 7. The 7 is in the remainder class 3 with respect to 4. So there can be no representation of 14 as the sum of two square numbers.

98 = 2 x 7 x 7. Here, too, it is true that 7 is in the remainder class 3 with respect to 4, but it is present twice in the prime factorization, so there can be a representation of 98 as the sum of two square numbers, namely 49 + 49.

Conversely, Fermat found the so-called two-squares theorem that every prime number for which the following applies: can be represented as the sum of two square numbers. This finding was used by mathematician Carl Gustav Jacob Jacobi to prove the theorem: ${\displaystyle p}$ ${\displaystyle p\equiv 1\mod 4}$

Any natural number can be represented as the sum of two squares if and only if all appear in even multiples in the prime factorization . ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle p\equiv 3\mod 4}$

The German mathematician Edmund Landau showed that the number of such numbers, which 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 necessary to any display any natural number as a sum of squares. The four-squares theorem above answers this question.

Relation to Euler's four-squares theorem

Has one with

${\displaystyle n_{1}=a_{1}^{2}+b_{1}^{2}+c_{1}^{2}+d_{1}^{2}}$    and    ${\displaystyle \qquad n_{2}=a_{2}^{2}+b_{2}^{2}+c_{2}^{2}+d_{2}^{2}}$

the representations of two numbers n ₁ and n ₂ as the sum of 4 squares, then one has over the quaternions

${\displaystyle x_{i}=a_{i}+b_{i}\cdot \mathrm {i} +c_{i}\cdot \mathrm {j} +d_{i}\cdot \mathrm {k} }$    and the equation    ${\displaystyle \qquad |x_{1}|^{2}\cdot |x_{2}|^{2}=|x_{1}x_{2}|^{2}}$

a representation of the product as a sum of 4 squares:

${\displaystyle n_{1}n_{2}=(a_{1}^{2}+b_{1}^{2}+c_{1}^{2}+d_{1}^{2})(a_{2}^{2}+b_{2}^{2}+c_{2}^{2}+d_{2}^{2})}$

${\displaystyle {}=(a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2}-d_{1}d_{2})^{2}}$

${\displaystyle {}+{}(a_{1}b_{2}+b_{1}a_{2}+c_{1}d_{2}-d_{1}c_{2})^{2}}$

${\displaystyle {}+{}(a_{1}c_{2}-b_{1}d_{2}+c_{1}a_{2}+d_{1}b_{2})^{2}}$

${\displaystyle {}+{}(a_{1}d_{2}+b_{1}c_{2}-c_{1}b_{2}+d_{1}a_{2})^{2}}$

Leonhard Euler had already discovered this identity in 1748, it is known as Euler's four-squares theorem . With this theorem he reduced the proof of the theorem that every number can be written as the sum of four square numbers to prime numbers. If prime numbers can be represented as sums of four squares, so are products of prime numbers; so are all natural numbers, since they are products of prime numbers.

Related problems and results

In 1798, Adrien-Marie Legendre dealt with the related question of representing the sums of natural numbers by at most three square numbers. He found and formulated that a natural number can always be composed of three or fewer square numbers if it is not of the form with integers . This theorem is also called the three-squares theorem . ${\displaystyle 4^{i}(8j+7)}$ ${\displaystyle i,j\geq 0}$

A loophole in Legendre's proof was later closed by Carl Friedrich Gauß , which is why it is also known as Gauß's theorem. Peter Gustav Lejeune Dirichlet and Edmund Landau found simplifications to the proof.

The three-square theorem draws not least the well-known (and already by de Pierre Fermat assumed) sentence after that every natural number as a sum of at most three triangular numbers can be represented .

As an extension of the question on which the four-squares theorem is based, Waring's problem deals with the question of whether there is   a number for every exponent so that every natural number can be written as the sum of at most -th powers, and the question that follows which way to find the smallest possible of these numbers . David Hilbert proved in 1909 that these always exist . ${\displaystyle k=2,\,3,\dotsc }$ ${\displaystyle g_{k}}$ ${\displaystyle g_{k}}$  ${\displaystyle k}$ ${\displaystyle g_{k}}$ ${\displaystyle g_{k}}$

Number of presentations

When calculating the respective number of representations of a natural number as a sum of four square numbers , the sign of the squared whole numbers and their order can be taken into account. ${\displaystyle n\in \mathbb {N} ^{*}}$

For example, the sum of four squares is shown for ${\displaystyle n=7}$

${\displaystyle 7=1^{2}+1^{2}+1^{2}+2^{2}=...=(-2)^{2}+(-1)^{2}+(-1)^{2}+(-1)^{2}}$

with the permutations of the tuples and total representations. ${\displaystyle (1,1,1,2),(-1,1,1,2),(-1,-1,1,2),(-1,-1,-1,2),(-2,1,1,1),(-2,-1,1,1),(-2,-1,-1,1)}$ ${\displaystyle (-2,-1,-1,-1)}$ ${\displaystyle r_{4}(7)=64}$

Jacobi's theorem provides a formula for the number of such representations .

See also

• Waring's problem
• Lipschitz quaternions
• Hurwitz quaternions
• Sum of squares function
• Two-squares set , three-squares set

literature

• Peter Bundschuh : Introduction to Number Theory . 5th edition, Springer-Verlag, 2002, ISBN 3-540-43579-4 , pp. 154-167.
• Otto Forster : Algorithmic Number Theory . Springer-Verlag, 1996, ISBN 978-3-663-09240-7 (print) 978-3-663-09239-1 (online), pp. 228-237
• Wacław Sierpiński : Elementary Theory of Numbers . Chapter XI: Represantations of Natural Numbers as Sums of Non-Negative kth Powers (=  North-Holland Mathematical Library . Volume 31 ). 2nd revised and expanded edition. North-Holland (inter alia), Amsterdam (inter alia) 1988, ISBN 0-444-86662-0 , pp. 378 ff . ( MR0930670 ).

Individual evidence

1. ^ P. 421 in John Stillwell: Mathematics and its history . 3. Edition. Springer, New York 2010, ISBN 978-1-4419-6052-8 , doi : 10.1007 / 978-1-4419-6053-5 .
2. ^ P. 423 in John Stillwell: Mathematics and its history . 3. Edition. Springer, New York 2010, ISBN 978-1-4419-6052-8 , doi : 10.1007 / 978-1-4419-6053-5 .
3. ↑ See letter from Leonhard Euler to Christian Goldbach (May 4, 1748 / April 12, 1749).
4. ↑ See Adrien-Marie Legendre: Essai sur la theory des Nombres . 2nd Edition. Paris 1808, pp. 293–339 ( Théorie des Nombres considérés comme décomposables en trois quarrés ).
5. ^ Wacław Sierpiński: Elementary Theory of Numbers 1988, pp. 391-392
6. ↑ David Hilbert: Proof of the representability of the whole numbers by a fixed number of n-th powers (Waring's problem). In: Mathematische Annalen , 67, 1909, pp. 281-300. Cf. Erhard Schmidt: On Hilbert's proofs of Waring's theorem. (From a letter addressed to Mr. Hilbert.) In: Mathematische Annalen , 74, 1913, No. 2, pp. 271–274.