Your friend and mine,
Leonhard Euler conjectured a generalization of this theorem, stating something to the effect of, "No
nth power is the sum of less than
n nth powers." That is, x^3 cannot be expressed as y^3 + z^3, nor can x^4 be expressed as y^4 + z^4 + w^4, etc (with x, y, z and w being
whole numbers).
This generalization was assumed to be true for a long while, as no counterexample could be found. In 1968, however, Leon Lander and Thomas Parkin, running a computer program they'd written to list out fifth powers that were the sum of five smaller fifth powers, accidentally stumbled upon 144^5 = 27^5 + 84^5 + 110^5 + 133^5 + 0^5. Apparently whichever one of them wrote the program used a greater than or equal to operator instead of a greater than operator in a for loop, or some such. It took 19 years to disprove the Euler's conjecture for n = 4; in 1987, the mathematician N.J. Elkies discovered that 20615673^4 = 2682440^4 + 15365639^4 + 18796760^4.
(And no, I did not just make up that set of numbers)