How are Pythagoras' theorem and Fermat's last theorem related?

Answer 1

Pythagoras' theorem proves that if you draw a square on the longest side (the hypotenuse) of a right-angled triangle, its area is the same as the areas of the squares drawn on the two shorter sides, added together. See 'Pythagoras' theorem' under 'Sources and related links' below.

Pythagoras' theorem holds for any right-angled triangle. But of special interest to Fermat were right-angled triangles where all the three sides were whole number lengths. These special lengths are known as Pythagorean triples.

Here are some Pythagorean triples:-

(3,4,5) (5, 12, 13) (7, 24, 25) (8, 15, 17)

In each case, the square of each of the smaller numbers is equal to the square of the largest number.

Fermat said that if instead of constructing squares (two dimensional figures) on the sides of right-angled triangles, you constructed cubes (three dimensional analogs of squares), or hypercubes (four dimensional analogs) or higher dimensional cube-analogs, there are no equivalents to the Pythagorean triples. In other words, there are no whole number values for 3, 4 or more dimensional analogs of the square.