Fermat's Polygonal Number Theorem

In 1638, Fermat proposed that every Positive Integer is a sum of at most three Triangular Numbers, four Square Numbers, five Pentagonal Numbers, and $n$ $n$-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

$$** E\Upsilon RHKA\qquad {\it num}=\Delta+\Delta+\Delta.$$

This case is equivalent to the statement that every number of the form $8m+3$ 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 $4^b(8m+7)$ for $b\geq 0$, 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.

See also Fifteen Theorem, Vinogradov's Theorem, Lagrange's Four-Square Theorem, Waring's Problem

References

Cassels, J. W. S. Rational Quadratic Forms. New York: Academic Press, 1978.

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.

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.