This post concerns the uniqueness of the nontrivial solution to (*) 1^2 + 2^2 + 3^3 + . . . + n^2 = K^2 (namely, n = 24 and K = 70), and is largely based on "The square pyramid puzzle" by W.S. Anglin, American Mathematical Monthly, Feb. 1990. The problem seems to have originated with an 1875 challenge from Edouard Lucas to prove that "A square pyramid of cannonballs contains a square number of cannonballs only when it has 24 cannonballs along its base." (He was evidently not counting the trivial solution of n = K = 1.) In 1876 a flawed proof was pubished by M. Moret-Blanc. In 1877 a flawed proof was pubished by Lucas. In 1918 -- 43 years after the problem was posed -- apparently the first valid proof (14 pages) was published, by G.N. Watson, using an extended theory of Jacobi elliptic functions. More proofs appeared in 1952, 1966, and 1975, but not until 1985 did someone (De Gang Ma) publish a proof accessible at the undergraduate level. In a simplification of De Gang Ma's method, in this article Anglin uses 3 lemmas to cover the case of n even, and 7 lemmas (and an entirely different strategy) to cover the case of n odd -- and draws on the theory of Pell's equation and quadratic reciprocity. The arguments involve considering a large number of cases and strike me as quite ad hoc, (although in total they cover only about 3 1/2 pages). My question to math-fun is this: Can this problem really be as difficult as this history would make it seem? Is it possible that considering the equation (*) modulo just the right primes would lead to an efficient proof? Or perhaps there could be a geometric argument? --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele