Legendre's three-square theorem

In mathematics, Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers

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

if and only if n is not of the form n = 4^a(8b + 7) for integers a and b.

The first numbers that cannot be expressed as the sum of three squares (i.e. numbers that can be expressed as n = 4^a(8b + 7)) are

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

History

Pierre de Fermat gave a criterion for numbers of the form 3a+1 to be a sum of 3 squares essentially equivalent to Legendre's theorem, but did not provide a proof. N. Beguelin noticed in 1774[1] that every positive integer which is neither of the form 8n + 7, nor of the form 4n, is the sum of three squares, but did not provide a satisfactory proof.[2] In 1796 Gauss proved his Eureka theorem that every positive integer n is the sum of 3 triangular numbers; this is trivially equivalent to the fact that 8n+3 is a sum of 3 squares. In 1797 or 1798 A.-M. Legendre obtained the first proof of his 3 square theorem.[3] In 1813, A. L. Cauchy noted[4] that Legendre's theorem is equivalent to the statement in the introduction above. Previously, in 1801, C. F. Gauss had obtained a more general result,[5] containing Legendre theorem of 1797-8 as a corollary. In particular, Gauss counted the number of solutions of the expression of an integer as a sum of three squares, and this is a generalisation of yet another result of Legendre,[6] whose proof is incomplete. This last fact appears to be the reason for later incorrect claims according to which Legendre's proof of the three-square theorem was defective and had to be completed by Gauss.[7]

With Lagrange's four-square theorem and the two-square theorem of Girard, Fermat and Euler, the Waring's problem for k = 2 is entirely solved.

Proofs

The "only if" of the theorem is simply because modulo 8, every square is congruent to 0, 1 or 4. There are several proofs of the converse (besides Legendre's proof). One of them is due to J. P. G. L. Dirichlet in 1850, and has become classical.[8] It requires three main lemmas:

Relationship to the four-square theorem

This theorem can be used to prove Lagrange's four-square theorem, which states that all natural numbers can be written as a sum of four squares. Gauss[9] pointed out that the four squares theorem follows easily from the fact that any positive integer that is 1 or 2 mod 4 is a sum of 3 squares, because any positive integer not divisible by 4 can be reduced to this form by subtracting 0 or 1 from it. However, proving the three-square theorem is considerably more difficult than a direct proof of the four-square theorem that does not use the three-square theorem. Indeed, the four-square theorem was proved earlier, in 1770.

See also

Notes

  1. ↑ Nouveaux Mémoires de l'Académie de Berlin (1774, publ. 1776), p. 313-369.
  2. ↑ Leonard Eugene Dickson, 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. 202 et 398-399.
  4. ↑ A. L. Cauchy, Mém. Sci. Math. Phys. de l'Institut de France, (1) 14 (1813-1815), 177.
  5. ↑ C. F. Gauss, Disquisitiones Arithmeticae, Art. 291 et 292.
  6. ↑ A.-M. Legendre, Hist. et Mém. Acad. Roy. Sci. Paris, 1785, p. 514-515.
  7. ↑ See for instance: Elena Deza and M. Deza. Figurate numbers. World Scientific 2011, p.314
  8. ↑ See for instance vol. I, parts I, II and III of : E. 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
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