The Pythagorean Theorem - Educational Outreach

An example of a Diophantine equation is the Pythagorean2 2 2relationship x y z where the solutions x , y,z arerestricted to integers. We examined the problem of finding2 2 2integer solutions to x y z in Section 4.2 when westudied Pythagorean Triples. Likewise, Diophantus himselfexamined Pythagorean Triples and wrote about them in hisArithmetica. The subject of Pythagorean Triples was thefocus of Fermat’s studies sometime in 1637 (as the storygoes) when he pondered a natural extension.Fermat’s Fundamental QuestionSince many sets of three integers x , y,z exist wherex2 y2 z2, could it be that there exist a set of threeintegers x , y,z such thatx4 y4 z4, etc.x3 3 3 y z ? What aboutFor the cubic relationship, one can cometantalizingly close as the following five examples show:633 371 138135 833 3791 8122676 91383331 1443 3230 1721 1010333131 37533x31 y.3 zBut as we say in present times, almost does not countexcept when one is tossing horseshoes or hand grenades.Fermat must have quickly come to the realization that3 3 3 4 4 4x y z , x y z and companions constituted aseries of Diophantine impossibilities, for he formulated thefollowing in the margin of his copy of Arithmetica.3131