Fermat's Enigma

by Simon Singh

At the heart of his proof of Fermat, Andrew had proved an idea known as the Taniyama-Shimura conjecture, which created a new bridge between wildly different mathematical worlds.

History was littered with false claims, and much as I wished that he would be the exception, it was hard to imagine Wiles as anything but another headstone in that mathematical graveyard.

The Last Theorem is at the heart of an intriguing saga of courage, skulduggery, cunning, and tragedy, involving all the greatest heroes of mathematics.

Pythagoras, the teacher, paid his student three oboli for each lesson he attended

Each member of the school was forced to swear an oath never to reveal to the outside world any of their mathematical discoveries.

Even after Pythagoras’s death a member of the Brotherhood was drowned for breaking his oath—he publicly announced the discovery of a new regular solid, the dodecahedron, constructed from twelve regular pentagons.

The Brotherhood was effectively a religious community, and one of the idols they worshiped was Number.

the sum of a number’s divisors is greater than the number itself, it is called an “excessive” number.

when the sum of a number’s divisors is less than the number itself, it is called “defective.”

those whose divisors add up exactly to the number itself, and these are the perfect numbers.

St. Augustine observed that 6 was not perfect because God chose it, but rather that the perfection was inherent in the nature of the number:

The third perfect number is 496, the fourth is 8,128, the fifth is 33,550,336, and the sixth is 8,589,869,056.

perfect numbers are always the sum of a series of consecutive counting numbers.

Pythagoras was entertained by perfect numbers, but he was not satisfied with merely collecting these special numbers; instead he desired to discover their deeper significance.

All these powers of 2 only just fail to be perfect, because the sum of their divisors always adds up to one less than the number itself. This makes them only slightly defective:

Euclid discovered that perfect numbers are always the multiple of two numbers, one of which is a power of 2 and the other being the next power of 2 minus 1.

although there are plenty of numbers whose divisors add up to one less than the number itself, that is to say only slightly defective, there appear to be no numbers that are slightly excessive.

two and a half thousand years later, mathematicians are still unable to prove that no slightly excessive numbers

Pythagoras had uncovered for the first time the mathematical rule that governs a physical phenomenon and demonstrated that there was a fundamental relationship between mathematics and science.

Pythagoras’s proof is irrefutable. It shows that his theorem holds true for every right-angled triangle in the universe. The discovery was so momentous that one hundred oxen were sacrificed as an act of gratitude to the gods.

Its significance was twofold. First, it developed the idea of proof.

The second consequence of Pythagoras’s theorem is that it ties the abstract mathematical method to something tangible.

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.

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.

This was the Pythagoreans’ greatest contribution to civilization—a way of achieving truth that is beyond the fallibility of human judgment.

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.

Pythagorean triples are combinations of three whole numbers that perfectly fit Pythagoras’s equation:

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.

in E. T. Bell’s The Last Problem,

Bell’s book described the existence of a mathematical monster.

finding whole number solutions to the sister equation appears to be impossible.

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.

Finding three numbers that fit the cubed equation perfectly seems to be impossible.

if the power is changed from 3 (cubed) to any higher number n (i.e., 4, 5, 6,…), then finding a solution seems equally impossible.

In fact, the great seventeenth-century Frenchman Pierre de Fermat made the astonishing claim that the reason why nobody could find any solutions was that no solutions

Fermat’s Last Theorem, as it is known, stated that xn + yn = zn has no whole number solutions for

As each generation failed, the next became even more frustrated and determined. In 1742, almost a century after Fermat’s death, the Swiss mathematician Leonhard Euler asked his friend Clêrot to search Fermat’s house in case some vital scrap of paper still remained. No clues were ever found as to what Fermat’s proof might have been.

Wiles wrote up the statement of Fermat’s Last Theorem, turned toward the audience, and said modestly: “I think I’ll stop here.”

the only institute in Europe to actively encourage mathematicians was Oxford University, which had established the Savilian Chair of Geometry in 1619.

Pascal’s interest in the subject had been sparked by a professional Parisian gambler, Antoine Gombaud, the Chevalier de Méré, who had posed a problem that concerned a game of chance called points.

Pascal was even convinced that he could use his theories to justify a belief in God.

Therefore, according to Pascal’s definition, religion was a game of infinite excitement and one worth playing, because multiplying an infinite prize by a finite probability results in infinity.

Newton wrote that he developed his calculus based on “Monsieur Fermat’s method of drawing tangents.”

Fermat’s greatest love was for a subject that is largely useless—the theory of numbers. Fermat was driven by an obsession to understand the properties of and the relationships between numbers. This is the purest and most ancient form of mathematics, and Fermat was building on a body of knowledge that had been handed down to him from Pythagoras.

The first head of the mathematics department was none other than Euclid.

In particular Euclid exploited a logical weapon known as reductio ad absurdum, or proof by contradiction.

“Reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.”

One of Euclid’s most famous proofs by contradiction established the existence of so-called irrational numbers. It is suspected that irrational numbers were originally discovered by the Pythagorean Brotherhood centuries earlier, but the concept was so abhorrent to Pythagoras that he denied their existence.

An irrational number is a number that is neither a whole number nor a fraction, and this is what made it so horrific to Pythagoras.

A recurring decimal such as 0.111111…is in fact a fairly straightforward number,