Three squares set

The three-square theorem of Legendre is a mathematical theorem from the theory of numbers , it reads:

A natural number can then be the sum of three square numbers ${\displaystyle n}$

${\displaystyle n=x^{2}+y^{2}+z^{2}}$

be written, if not of the form ${\displaystyle n}$

${\displaystyle n=4^{a}(8b+7)}$

with natural numbers and is. ${\displaystyle a}$ ${\displaystyle b}$

The first numbers that cannot be written as the sum of three square numbers are

7, 15, 23, 28, 31, 39, 47, 55, 60, 63, 71 ... sequence A004215 in OEIS .

Historical remarks

Pierre de Fermat found a criterion for when a natural number is of the form sum of three squares, which is essentially equivalent to Legendre's theorem , but he gave no proof. Nikolaus von Béguelin observed in 1774 that any positive number that has neither the shape nor the form is the sum of three squares, again without satisfactory evidence. In 1796 Carl Friedrich Gauss proved the so-called Eureka theorem , according to which every natural number is the sum of three triangular numbers ; this is equivalent to a number of the form being the sum of three squares. Adrien-Marie Legendre succeeded in proving the three-squares theorem for the first time in 1797 or 1798. In 1813 Augustin Louis Cauchy remarked that Legendre's theorem is equivalent to the formulation given in the introduction. Earlier, in 1801, Gauss had derived a more general result that contained Legendre's theorem as a corollary . In particular, Gauss counted the number of possible representations of a number as the sum of three squares, thus generalizing another result of Legendre, whose proof was incomplete. This latter circumstance seems to be the cause of later false claims that Legendre's three-squares theorem was flawed and was only completed by Gauss. ${\displaystyle n=3a+1}$ ${\displaystyle 8a+7}$ ${\displaystyle 4a}$ ${\displaystyle 8a+3}$

proofs

The line of proof that a sum of three squares cannot have the shape follows very easily from the fact that a square modulo 8 is congruent to 0, 1 or 4. In addition to Legendre's proof, there are several more proofs of the reverse. One goes back to JPGL Dirichlet from 1850, which is now considered classic. He uses three essential ingredients: ${\displaystyle n=4^{a}(8b+7)}$

• the quadratic reciprocity law
• the Dirichlet prime number theorem in arithmetic progressions
• the equivalence class of the trivial ternary quadratic form

Relationship to the four-squares theorem

The three-square theorem can be used, the four-square theorem of Lagrange to prove that says that every natural number is the sum of four squares. Gauss pointed out that the four-squares theorem easily follows from the fact that any number congruent to 1 or 2 modulo 4 is the sum of three squares, because any number not divisible by 4 can be calculated by subtracting 0 or 1 can be brought to this form. However, a direct proof of the four squares theorem is considerably easier than taking this detour via the three squares theorem. In fact, the four-squares theorem was proven earlier, namely in 1770.

See also

• Fermat's two-squares theorem
• Sum of squares function
• Pythagorean quadruple

Left

• Proof of the three-squares theorem on maths fun

Individual evidence

1. ^ Nouveaux Mémoires de l'Académie de Berlin (1774, published 1776), pages 313-369.
2. ^ Leonard Eugene Dickson : History of the theory of numbers , Volume II, p. 15 (Carnegie Institute of Washington 1919; AMS Chelsea Publ., 1992, reprint).
3. ↑ A.-M. Legendre: Essai sur la théorie des nombres , Paris, An VI (1797–1798), page 202 and pages 398-399
4. ↑ AL Cauchy: Mém. Sci. Math. Phys. de l'Institut de France , (1) 14 (1813-1815), page 177
5. ↑ CF Gauss, Disquisitiones Arithmeticae , Art. 291 et 292.
6. ↑ A.-M. Legendre, Hist. et Mém. Acad. Roy. Sci. Paris , 1785, pp. 514-515.
7. ↑ See for example Elena Deza and M. Deza: Figurate numbers . World Scientific 2011, page 314 [1]
8. ↑ See, for example, E. Landau : Lectures on Number Theory , New York, Chelsea, 1927, Volume I, Parts I, II and III
9. ↑ CF Gauss, Disquisitiones Arithmeticae , Art. Art 293.