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
if and only if n is not of the form 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 ) 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
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" (PDF). September 25, 1654. Archived (PDF) from the original on July 5, 2017.
- ^ 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.
- ISBN 0-300-09473-6