Squares set

In mathematics , the squares theorem indicates for which natural numbers n the product of two sums of n squared real numbers decays into a sum of likewise n squares of numbers that are bilinear forms of the former. Since 1818 it is known that this for n is possible = 1,2,4 or 8 and the composition rate of Adolf Hurwitz from 1898 says that this is also the only n are. The norms || · || of real and complex numbers, quaternions and octonions satisfy the relation || a || || b || = || from ||, from which the known compositions can be constructed. As a direct consequence of the identities it follows that the set of sums of n square numbers is closed with respect to the multiplication in the mentioned cases.

For n = 2 he was already known to Diophantos of Alexandria . Adrien-Marie Legendre was the first to find out that it does not hold for n = 3 (in his textbook on number theory). Leonhard Euler proved the case n = 4 in a letter to Goldbach in 1748. The case n = 8 was found by John T. Graves in 1844 in connection with the theory of the octaves he introduced (and by Arthur Cayley in 1845).

statement

There are bilinear forms only for n = 1, 2, 4 or 8

${\displaystyle c_{i}=\sum _{j,k=1}^{n}\gamma _{ijk}a_{j}b_{k},\quad \gamma _{ijk}\in \mathbb {R} }$

for i = 1, ..., n , so that for all real numbers a ₁ , ..., a _n , b ₁ , ..., b _n applies:

${\displaystyle \left(\sum _{j=1}^{n}a_{j}^{2}\right)\left(\sum _{k=1}^{n}b_{k}^{2}\right)=\sum _{i=1}^{n}c_{i}^{2}}$

proof

After the composition set of quadratic forms of Adolf Hurwitz is for n bilinear functions z ₁ , z ₂ , ..., z _n of 2 n independent real variable x ₁ , x ₂ , ..., x _n and y ₁ , y ₂ , ..., y _n the equation

${\displaystyle (x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2})(y_{1}^{2}+y_{2}^{2}+\cdots +y_{n}^{2})=z_{1}^{2}+z_{2}^{2}+\cdots +z_{n}^{2}}$

only solvable if n = 1, 2, 4 or 8. On the other hand, the compositions listed below for these n were already known in Hurwitz's time , which completes the proof:

"With this evidence, the old controversial question of whether the well-known product formulas for sums of 2, 4 and 8 squares can be extended to sums of more than 8 squares is finally decided in a negative sense."

- Adolf Hurwitz (1898)

Case n = 1

The statement for n = 1 is written out

${\displaystyle a^{2}b^{2}=(ab)^{2}}$

which applies to all . ${\displaystyle a,b\in \mathbb {R} }$

Brahmagupta identity

→ Main article : Brahmagupta identity

As early as 628 AD, the Indian mathematician and astronomer Brahmagupta proved an identity, the two-squares theorem

${\displaystyle {\begin{aligned}(a^{2}+b^{2})(u^{2}+v^{2})=&(au-bv)^{2}+(av+bu)^{2}\\=&(au+bv)^{2}+(av-bu)^{2}\end{aligned}}}$

contains as a special case. This can be confirmed by multiplying it out, but it also results from the relation | z | ² | w | ² = | zw | ² for complex numbers z = a ± i b , w = u ± i v and the imaginary unit i² = - 1.

On closer inspection, the formula for the case n = 1 appears in each quadrant:

${\displaystyle a^{2}u^{2}=(au)^{2},\;a^{2}v^{2}=(av)^{2},\;b^{2}u^{2}=(bu)^{2},\;b^{2}v^{2}=(bv)^{2}.}$

This abnormality is similarly encountered in the following cases.

Euler's four-squares theorem

Leonhard Euler has the relation in 1748

${\displaystyle {\begin{array}{c}[(a_{1}^{2}+a_{2}^{2})+(a_{3}^{2}+a_{4}^{2})][(b_{1}^{2}+b_{2}^{2})+(b_{3}^{2}+b_{4}^{2})]=\\\left[(a_{1}b_{1}-a_{2}b_{2})+(-a_{3}b_{3}-a_{4}b_{4})\right]^{2}\\+[(a_{1}b_{2}+a_{2}b_{1})+(a_{3}b_{4}-a_{4}b_{3})]^{2}\\+[(a_{1}b_{3}-a_{2}b_{4})+(a_{3}b_{1}+a_{4}b_{2})]^{2}\\+[(a_{1}b_{4}+a_{2}b_{3})+(-a_{3}b_{2}+a_{4}b_{1})]^{2}\end{array}}}$

discovered, also known as the general four-squares theorem . Today it results from the product rule for the norms of quaternions || a || || b || = || ab ||, see Lagrange's four-squares theorem , which Joseph Louis Lagrange derived from Euler's relation in 1770.

As announced, the sentence for n = 2 appears here for each quadrant , for example

${\displaystyle (a_{3}^{2}+a_{4}^{2})(b_{1}^{2}+b_{2}^{2})=(a_{3}b_{1}+a_{4}b_{2})^{2}+(-a_{3}b_{2}+a_{4}b_{1})^{2}}$

Degen's eight squares set

Degen's eight-squares theorem shows that the product of two numbers that are a sum of eight squares are themselves the sum of eight squares:

${\displaystyle {\begin{array}{c}(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}+a_{7}^{2}+a_{8}^{2})(b_{1}^{2}+b_{2}^{2}+b_{3}^{2}+b_{4}^{2}+b_{5}^{2}+b_{6}^{2}+b_{7}^{2}+b_{8}^{2})=\\(a_{1}b_{1}-a_{2}b_{2}-a_{3}b_{3}-a_{4}b_{4}-a_{5}b_{5}-a_{6}b_{6}-a_{7}b_{7}-a_{8}b_{8})^{2}+\\(a_{1}b_{2}+a_{2}b_{1}+a_{3}b_{4}-a_{4}b_{3}+a_{5}b_{6}-a_{6}b_{5}-a_{7}b_{8}+a_{8}b_{7})^{2}+\\(a_{1}b_{3}-a_{2}b_{4}+a_{3}b_{1}+a_{4}b_{2}+a_{5}b_{7}+a_{6}b_{8}-a_{7}b_{5}-a_{8}b_{6})^{2}+\\(a_{1}b_{4}+a_{2}b_{3}-a_{3}b_{2}+a_{4}b_{1}+a_{5}b_{8}-a_{6}b_{7}+a_{7}b_{6}-a_{8}b_{5})^{2}+\\(a_{1}b_{5}-a_{2}b_{6}-a_{3}b_{7}-a_{4}b_{8}+a_{5}b_{1}+a_{6}b_{2}+a_{7}b_{3}+a_{8}b_{4})^{2}+\\(a_{1}b_{6}+a_{2}b_{5}-a_{3}b_{8}+a_{4}b_{7}-a_{5}b_{2}+a_{6}b_{1}-a_{7}b_{4}+a_{8}b_{3})^{2}+\\(a_{1}b_{7}+a_{2}b_{8}+a_{3}b_{5}-a_{4}b_{6}-a_{5}b_{3}+a_{6}b_{4}+a_{7}b_{1}-a_{8}b_{2})^{2}+\\(a_{1}b_{8}-a_{2}b_{7}+a_{3}b_{6}+a_{4}b_{5}-a_{5}b_{4}-a_{6}b_{3}+a_{7}b_{2}+a_{8}b_{1})^{2}\end{array}}}$

This relation was found in 1818 by Carl Ferdinand Degen , who, however, mistakenly believed that he could generalize it to 2 ^m squares, which is what John Thomas Graves (1843) believed. The latter and Arthur Cayley (1845) derived the relation from the octonions independently of each other and of Degen . For those - as with the quaternions - || a || || b || = || from ||, from which the above relation follows through calculation.

In this equation, each quadrant represents a version of Euler's four-squares theorem, for example

${\displaystyle {\begin{array}{c}(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2})(b_{1}^{2}+b_{2}^{2}+b_{3}^{2}+b_{4}^{2})=\\(a_{1}b_{1}-a_{2}b_{2}-a_{3}b_{3}-a_{4}b_{4})^{2}+\\(a_{1}b_{2}+a_{2}b_{1}+a_{3}b_{4}-a_{4}b_{3})^{2}+\\(a_{1}b_{3}-a_{2}b_{4}+a_{3}b_{1}+a_{4}b_{2})^{2}+\\(a_{1}b_{4}+a_{2}b_{3}-a_{3}b_{2}+a_{4}b_{1})^{2}\end{array}}}$

literature

• Guido Walz (Ed.): Lexicon of Mathematics . tape 4 (Moo-Sch). Springer Spektrum Verlag, Mannheim 2017, ISBN 978-3-662-53499-1 , doi : 10.1007 / 978-3-662-53500-4 .

Individual evidence

1. ↑ It was discovered by C. Degen in 1818. See Ebbinghaus u. a., Numbers, Springer 1983, p. 175 (on the eight squares sentence and in other parts of the book on the other cases).
2. ↑ Adolf Hurwitz : About the composition of the square forms of any number of variables . In: News from the k. Society of Sciences in Göttingen, mathematical-physical class . 1898, p. 309–316 ( Computer Science University of Toronto [PDF; accessed June 18, 2017]).
3. ↑ Guido Walz (Ed.): Lexicon of Mathematics . tape 5 (Sed to Zyl). Springer Spektrum Verlag, Mannheim 2017, ISBN 978-3-662-53505-9 , doi : 10.1007 / 978-3-662-53506-6 .
4. ^ Klaus Lamotke: Numbers . Springer-Verlag, 2013, ISBN 978-3-642-58155-7 ( limited preview in Google book search).