Fermat’s Proof

Mathematics most famous marginal note is Fermat’s assertion of what is now called Fermat’s Last Theorem. This was made in his edition of Arithmetica by Diophantus, which had recently been translated from Greek to Latin, the common language of science and all forms of learning. Fermat’s note was also in Latin, and translates roughly as follows: “It is impossible for a cube and a cube to give a cube, or a fourth power and a fourth power to give a fourth power, or, more generally, for any power of degree greater than two to be the sum of two powers of the given degree. I have a truly marvelous proof for this, but this margin is too small to contain it.” In other words, x + y = z has no solution if n > 2 and if x, y, z are required to be positive integers. What is less known is that Fermat was able to fit a proof of a related result in the margin in a later section of this same book. The result in this other marginal note can be used to give a quick proof of the n = 4 case of Fermat’s Last Theorem. It is not known whether he proved the n = 4 case in this way, or saw the connection between the two marginal notes. However, it is believed that Fermat did have some sort of proof of the n = 4 case of “Fermat’s Last Theorem” since it can be proved with techniques well-known to Fermat, and since he claimed to be able to prove the n = 3 and n = 4 cases in several letters. In contrast, Fermat never publically claimed a proof of the general case (n > 4). The claim was confined to the above mentioned marginal note written apparently only for himself. The result that Fermat stated in this later marginal note is related to Pythagorean triples, right triangles with all three sides of integral length: