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
- [math]\displaystyle{ n = x^2 + y^2 + z^2 }[/math]
if and only if n is not of the form [math]\displaystyle{ n = 4^a(8b + 7) }[/math] for nonnegative 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 [math]\displaystyle{ n = 4^a(8b + 7) }[/math]) are
a b
|
0 | 1 | 2 |
|---|---|---|---|
| 0 | 7 | 28 | 112 |
| 1 | 15 | 60 | 240 |
| 2 | 23 | 92 | 368 |
| 3 | 31 | 124 | 496 |
| 4 | 39 | 156 | 624 |
| 5 | 47 | 188 | 752 |
| 6 | 55 | 220 | 880 |
| 7 | 63 | 252 | 1008 |
| 8 | 71 | 284 | 1136 |
| 9 | 79 | 316 | 1264 |
| 10 | 87 | 348 | 1392 |
| 11 | 95 | 380 | 1520 |
| 12 | 103 | 412 | 1648 |
| Unexpressible values up to 100 are in bold | |||
History
Pierre de Fermat gave a criterion for numbers of the form 8a + 1 and 8a + 3 to be sums of a square plus twice another square, but did not provide a proof.[1] N. Beguelin noticed in 1774[2] 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.[3] In 1796 Gauss proved his Eureka theorem that every positive integer n is the sum of 3 triangular numbers; this is equivalent to the fact that 8n + 3 is a sum of three squares. In 1797 or 1798 A.-M. Legendre obtained the first proof of his 3 square theorem.[4] In 1813, A. L. Cauchy noted[5] 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,[6] containing Legendre's 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,[7] 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.[8]
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.[9] It requires three main lemmas:
- the quadratic reciprocity law,
- Dirichlet's theorem on arithmetic progressions, and
- the equivalence class of the trivial ternary quadratic form.
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[10] 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
- Fermat's two-square theorem
- Sum of two squares theorem
- Legendre's equation
Notes
- ↑ "Fermat to Pascal". September 25, 1654. http://science.larouchepac.com/fermat/16540925%20Fermat%20to%20Pascal.pdf.
- ↑ Nouveaux Mémoires de l'Académie de Berlin (1774, publ. 1776), pp. 313–369.
- ↑ Leonard Eugene Dickson, 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. 202 and pp. 398–399.
- ↑ A. L. Cauchy, Mém. Sci. Math. Phys. de l'Institut de France, (1) 14 (1813–1815), 177.
- ↑ C. F. Gauss, Disquisitiones Arithmeticae, Art. 291 et 292.
- ↑ A.-M. Legendre, Hist. et Mém. Acad. Roy. Sci. Paris, 1785, pp. 514–515.
- ↑ See for instance: Elena Deza and M. Deza. Figurate numbers. World Scientific 2011, p. 314 [1]
- ↑ 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.
- ↑ Gauss, Carl Friedrich (1965), Disquisitiones Arithmeticae, Yale University Press, p. 342, section 293, ISBN 0-300-09473-6
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