Fermat’s Last Theorem. Simon Singh

a milestone in mathematics and one of the most important breakthroughs in the history of civilisation. Its significance was twofold. First, it developed the idea of proof. A proven mathematical result has a deeper truth than any other truth because it is the result of step-by-step logic. Although the philosopher Thales had already invented some primitive geometrical proofs, Pythagoras took the idea much further and was able to prove far more ingenious mathematical statements. The second consequence of Pythagoras’ theorem is that it ties the abstract mathematical method to something tangible. Pythagoras showed that the truth of mathematics could be applied to the scientific world and provide it with a logical foundation. Mathematics gives science a rigorous beginning and upon this infallible foundation scientists add inaccurate measurements and imperfect observations.

      An Infinity of Triples

      The Pythagorean Brotherhood invigorated mathematics with its zealous search for truth via proof. News of their success spread and yet the details of their discoveries remained a closely guarded secret. Many requested admission to the inner sanctum of knowledge, but only the most brilliant minds were accepted. One of those who was blackballed was a candidate by the name of Cylon. Cylon took exception to his humiliating rejection and twenty years later he took his revenge.

      During the sixty-seventh Olympiad (510 BC) there was a revolt in the nearby city of Sybaris. Telys, the victorious leader of the revolt, began a barbaric campaign of persecution against the supporters of the former government, which drove many of them to seek sanctuary in Croton. Telys demanded that the traitors be returned to Sybaris to suffer their due punishment, but Milo and Pythagoras persuaded the citizens of Croton to stand up to the tyrant and protect the refugees. Telys was furious and immediately gathered an army of 300,000 men and marched on Croton, where Milo defended the city with 100,000 armed citizens. After seventy days of war Milo’s supreme generalship led him to victory and as an act of retribution he turned the course of the river Crathis upon Sybaris to flood and destroy the city.

      Despite the end of the war, the city of Croton was still in turmoil because of arguments over what should be done with the spoils of war. Fearful that the lands would be given to the Pythagorean elite, the ordinary folk of Croton began to grumble. There had already been growing resentment among the masses because the secretive Brotherhood continued to withhold their discoveries, but nothing came of it until Cylon emerged as the voice of the people. Cylon preyed on the fear, paranoia and envy of the mob and led them on a mission to destroy the most brilliant school of mathematics the world had ever seen. Milo’s house and the adjoining school were surrounded, all the doors were locked and barred to prevent escape and then the burning began. Milo fought his way out of the inferno and fled, but Pythagoras, along with many of his disciples, was killed.

      Mathematics had lost its first great hero, but the Pythagorean spirit lived on. The numbers and their truths were immortal. Pythagoras had demonstrated that more than any other discipline mathematics is a subject which is not subjective. His disciples did not need their master to decide on the validity of a particular theory. A theory’s truth was independent of opinion. Instead the construction of mathematical logic had become the arbiter of truth. This was the Pythagoreans’ greatest contribution to civilisation – a way of achieving truth which is beyond the fallibility of human judgement.

      Following the death of their founder and the attack by Cylon, the Brotherhood left Croton for other cities in Magna Graecia, but the persecution continued and eventually many of them had to settle in foreign lands. This enforced migration encouraged the Pythagoreans to spread their mathematical gospel throughout the ancient world. Pythagoras’ disciples set up new schools and taught their students the method of logical proof. In addition to their proof of Pythagoras’ theorem, they also explained to the world the secret of finding so-called Pythagorean triples.

      

      Figure 3. Finding whole number solutions to Pythagoras’ equation can be thought of in terms of finding two squares which can be added together to form a third square. For example, a square made of 9 tiles can be added to a square of 16 tiles, and rearranged to form a third square made of 25 tiles.

      

      Pythagorean triples are combinations of three whole numbers which perfectly fit Pythagoras’ equation: x² + y² = z² For example, Pythagoras’ equation holds true if x = 3, y = 4 and z = 5:

      Another way to think of Pythagorean triples is in terms of rearranging squares. If one has a 3 × 3 square made of 9 tiles, and a 4 × 4 square made of 16 tiles, then all the tiles can be rearranged to form a 5 × 5 square made of 25 tiles, as shown in Figure 3.

      The Pythagoreans wanted to find other Pythagorean triples, other squares which could be added to form a third, larger square. Another Pythagorean triple is x = 5, y = 12 and z = 13:

      A larger Pythagorean triple is x = 99, y = 4,900 and z = 4,901. Pythagorean triples become rarer as the numbers increase, and finding them becomes harder and harder. To discover as many triples as possible the Pythgoreans invented a methodical way of finding them, and in so doing they also demonstrated that there are an infinite number of Pythagorean triples.

      From Pythagoras’ Theorem to Fermat’s Last Theorem

      Pythagoras’ theorem and its infinity of triples was discussed in E.T. Bell’s The Last Problem, the library book which caught the attention of the young Andrew Wiles. Although the Brotherhood had achieved an almost complete understanding of Pythagorean triples, Wiles soon discovered that this apparently innocent equation, x² + y² = z², has a darker side – Bell’s book described the existence of a mathematical monster.

      In Pythagoras’ equation the three numbers, x, y and z, are all squared (i.e. x² = x × x):

      However, the book described a sister equation in which x, y and z are all cubed (i.e. x³ = x × x × x). The so-called power of x in this equation is no longer 2, but rather 3:

      Finding whole number solutions, i.e. Pythagorean triples, to the original equation was relatively easy, but changing the power from ‘2’ to ‘3’ (the square to a cube) and finding whole number solutions to the sister equation appears to be impossible. Generations of mathematicians scribbling on notepads have failed to find numbers which fit the equation perfectly.

      

      Figure 4. Is it possible to add the building blocks from one cube to another cube, to form a third, larger cube? In this case a 6 × 6 × 6 cube added to an 8 × 8 × 8 cube does not have quite enough building blocks to form a 9 × 9 × 9 cube. There are 216 (6³) building blocks in the first cube, and 512 (8³) in the second. The total is 728 building blocks, which is 1 short of 9³.

      With the original ‘squared’ equation, the challenge was to rearrange the tiles in two squares to form a third, larger square. The ‘cubed’ version of the challenge is to rearrange two cubes made of building blocks, to form a third, larger cube. Apparently, no matter what cubes are chosen to begin with, when they are combined the result is either a complete cube with some extra blocks left over, or an incomplete cube. The nearest that