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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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V.10 Fermat’s Last Theorem

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 x2 + y2 = z2. 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 (m2n2)2 + (2mn)2 = (m2 + n2)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 xn + yn = zn for some power n ≥ 3? For ...

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