Many people, even if they are not mathematicians, are aware of the existence of *Pythagorean triples*: that is, triples of positive integers (*x*, *y*, *z*) such that *x*^{2} + *y*^{2} = *z*^{2}. These give us examples of right-angled triangles with integer side lengths, of which the best known is the “(3, 4, 5) triangle.” For any two integers *m* and *n*, we have that (*m*^{2}–*n*^{2})^{2} + (2*mn*)^{2} = (*m*^{2} + *n*^{2})^{2}, which gives us an infinite supply of Pythagorean triples, and in fact every Pythagorean triple is a multiple of a triple of this form.

FERMAT [VI.12] asked the very natural question of whether similar triples existed for higher powers: that is, could there be a solution in positive integers of the equation *x ^{n}* +

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