Legendre's Three Squares Theorem

Legendre's three-squares theorem states that a natural number can be represented by the sum of three squares of integers

n=x^{2}+y^{2}+z^{2}

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

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

Then and only if n is not representable in the form $n=4^{a}(8b+7)$ ${\ displaystyle n = 4 ^ {a} (8b + 7)}$ ${\ displaystyle n = 4 ^ {a} (8b + 7)}$ where a and b are integers.

In particular, numbers not representable by the sum of three squares and representable in the form $n=4^{a}(8b+7)$ ${\ displaystyle n = 4 ^ {a} (8b + 7)}$ ${\ displaystyle n = 4 ^ {a} (8b + 7)}$ are

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

Content

1 History
2 Evidence
3 Relationship with the four-square theorem
4 See also
5 notes

History

Pierre Fermat gave a criterion for the representability of numbers of the form $3n+1$ ${\ displaystyle 3n + 1}$ $3n+1$ the sum of three squares, but did not provide evidence. Nicholas de Begelin observed in 1774 ^[1] that any natural number not representable in the form $8b+7$ ${\ displaystyle 8b + 7}$ $8b+7$ and in uniform $4b$ ${\ displaystyle 4b}$ $4b$ there is a sum of not more than three squares, but did not provide satisfactory evidence. ^[2] In 1796, Gauss proved that any natural number is the sum of no more than three triangular numbers . It follows that $8n+3$ ${\ displaystyle 8n + 3}$ $8n+3$ the sum of no more than three squares. In 1797 or 1798, Legendre received the first proof of the three-square theorem. ^[3] In 1813, Cauchy remarked ^[4] that Legendre's theorem is equivalent to the above statement. Earlier in 1801, Gauss obtained a more general result, ^{[5] the} consequence of which was the Legendre theorem. In particular, Gauss calculated the number of solutions for the integer equation of three squares, and at the same time gave a generalization of another result of Legendre, ^[6] whose proof was incomplete. This probably led to erroneous claims that Legendre's evidence was incomplete and completed by Gauss. ^[7]

The Lagrange theorem on the sum of four squares and the three-square theorem give a complete solution to the Waring problem for k = 2.

Evidence

Proof that numbers $n=4^{a}(8b+7)$ ${\ displaystyle n = 4 ^ {a} (8b + 7)}$ $n=4^{a}(8b+7)$ are not representable by the sum of three squares is uncomplicated and it follows from the fact that any square modulo 8 is congruent to 0, 1 or 4.

For the remaining numbers to be represented by three squares, there are several proofs in addition to the proof of Legendre. The 1850 Dirichlet proof has become a classic. ^[8] It is based on three lemmas:

The quadratic law of reciprocity ,
Dirichlet's theorem on prime numbers in arithmetic progression ,
The equivalence class of a trivial three-membered quadratic form .

Relationship with the Four-Squares Theorem

Gauss noted ^[9] that the three-squares theorem makes it easy to prove the four-squares theorem. However, the proof of the three-squares theorem is much more complicated than the direct proof of the four-squares theorem, which was first proved in 1770.

Notes

↑ Nouveaux Mémoires de l'Académie de Berlin (1774, publ. 1776), pp. 313–369.
↑ Dixon, Leonard Eugene , History of the theory of numbers , vol. II, p. 15 (Carnegie Institute of Washington 1919; AMS Chelsea Publ., 1992, reprint).
↑ A.-M. Legendre, Essai sur la théorie des nombres , Paris, An VI (1797–1798), P. and pp. 398-399.
↑ AL Cauchy, Mém. Sci. Math. Phys. de l'Institut de France , (1) 14 (1813–1815), 177.
↑ CF Gauss, Disquisitiones Arithmeticae , Art. 291 et 292.
↑ A.-M. Legendre, Hist. et Mém. Acad. Roy. Sci. Paris , 1785, pp. 514-515.
↑ See, for example: Elena Deza and M. Deza. Figurate numbers . World Scientific 2011, p. 314 [1]
↑ vol. I, parts I, II and III of: Landau , Vorlesungen über Zahlentheorie , New York, Chelsea, 1927. Second edition translated into English by Jacob E. Goodman, Providence RH, Chelsea, 1958.
↑ Gauss, Carl Friedrich (1965), Disquisitiones Arithmeticae , Yale University Press, p. 342, section 293, ISBN 0-300-09473-6

[1] Nouveaux Mémoires de l'Académie de Berlin (1774, publ. 1776), pp. 313–369.

[2] Dixon, Leonard Eugene , History of the theory of numbers , vol. 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), P. and pp. 398-399.

[4] AL Cauchy, Mém. Sci. Math. Phys. de l'Institut de France , (1) 14 (1813–1815), 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, p. 314 [1]

[8] vol. I, parts I, II and III of: Landau , Vorlesungen über Zahlentheorie , New York, Chelsea, 1927. Second edition translated into English by Jacob E. Goodman, Providence RH, Chelsea, 1958.

[9] Gauss, Carl Friedrich (1965), Disquisitiones Arithmeticae , Yale University Press, p. 342, section 293, ISBN 0-300-09473-6