In 1638, Fermat proposed that every positive integer is a sum of at most three triangular numbers,
four square numbers, five pentagonal
numbers, and -polygonal numbers. Fermat
claimed to have a proof of this result, although Fermat's proof has never been found.
Gauss proved the triangular case, and noted the event in his diary on July 10, 1796,
with the notation
This case is equivalent to the statement that every number of the form is a sum of three oddsquares (Duke 1997). More specifically, a number is
a sum of three squaresiff
it is not of the form for , as first proved by Legendre in 1798.
Euler was unable to prove the square case of Fermat's theorem, but he left partial results which were subsequently used by Lagrange. The square case was finally proved
by Jacobi and independently by Lagrange in 1772. It is therefore sometimes known
as Lagrange's four-square theorem.
In 1813, Cauchy proved the proposition in its entirety.
Cassels, J. W. S. Rational Quadratic Forms. New York: Academic Press, 1978.Cauchy, A. "Démonstration
du théorème général de Fermat sur les nombres polygones."
In Oeuvres complètes d'Augustin Cauchy, Vol. VI (II Série).
Paris: Gauthier-Villars, pp. 320-353, 1905.Conway, J. H.;
Guy, R. K.; Schneeberger, W. A.; and Sloane, N. J. A. "The
Primary Pretenders." Acta Arith.78, 307-313, 1997.Duke,
W. "Some Old Problems and New Results about Quadratic Forms." Not. Amer.
Math. Soc.44, 190-196, 1997.Nathanson, M. B. "A
Short Proof of Cauchy's Polygonal Number Theorem." Proc. Amer. Math. Soc.9,
22-24, 1987.Savin, A. "Shape Numbers." Quantum11,
14-18, 2000.Shanks, D. Solved
and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 143-144,
1993.Smith, D. E. A
Source Book in Mathematics. New York: Dover, p. 91, 1984.
-polygonal numbers. Fermat
claimed to have a proof of this result, although Fermat's proof has never been found.
Gauss proved the triangular case, and noted the event in his diary on July 10, 1796,
with the notation
is a sum of three odd squares (Duke 1997). More specifically, a number is
a sum of three squares iff
it is not of the form 
for 
, as first proved by Legendre in 1798.
