Pythagoras's Theorem says that a² + b² = c² in right angle triangles. One such triangle has side lengths 3, 4 and 5. Another has sides 2291, 2700 and 3541. We'll begin this topic by working out a system to classify all of the triangles that have whole number length sides.

Fermat asked a more general question. Can we find whole number solutions to the very similar looking equation $a^3\; +\; b^3\; =\; c^3\; ?\; Or$ a^4\; +\; b^4\; =\; c^4\; ?\; Or$ a^312\; +\; b^312\; =\; c^312\; ?\; He\; declared,\; in\; the\; margin\; of\; a\; book,\; that\; the\; answer\; to\; these\; three\; questions,\; and\; any\; similar\; question\; where\; we\; replace\; the\; power\; of\; two\; in\; Pythagoras\text{'}s\; Theorem\; with\; a\; larger\; number,\; is\; No.\; He\; did\; not\; give\; the\; details\; of\; how\; he\; came\; to\; this\; conclusion,\; and\; one\; of\; the\; greatest\; mysteries\; in\; math\; was\; born.$$$

For the next several centuries, many great mathematicians (and even more lesser ones) worked at the problem - a problem with no motivation other than "Hey, we should be able to do this, why can't we?" - producing plenty of beautiful math along the way. The problem was finally laid to rest in the 1990s, using some of the most powerful mathematical developments of the twentieth century and linking in a deep way two seemingly unrelated branches of math.

http://cs.marlboro.edu/ courses/ fall2007/math_tour/ FLT
last modified Friday September 28 2007 1:42 pm EDT