Fermat's Enigma

by Simon Singh

This regularity, despite the fact that it continues to infinity, means that the decimal can be rewritten as a fraction.

The concept of an irrational number was a tremendous breakthrough. Mathematicians were

The nineteenth-century mathematician Leopold Kronecker said, “God made the integers; all the rest is the work of man.”

A beautiful feature of this random pattern is that it can be computed using an equation that is supremely regular:

Although knowing π to 39 decimal places is sufficient to calculate the circumference of the universe accurate to the radius of a hydrogen atom,

The current record is held by Yasumasa Kanada of the University of Tokyo, who calculated π to six billion decimal places in 1996.

Euclid dared to confront the issue of irrationality in the tenth volume of the Elements

he examined the square root of two, —the

Euclid used reductio ad absurdum and began by assuming that it could be written as a fraction. He then demonstrated that this hypothetical fraction could be simplified.

Euclid showed that his simplified fraction, which still was supposed to represent , could be simplified not just once but over and over again an infinite number of times without ever reducing to its simplest form. This is absurd because all fractions must eventually have a simplest form, and therefore the hypothetical fraction cannot exist.

For the first time, numbers had taken on a new and more abstract quality. Until this point in history all numbers could be expressed as whole numbers or fractions, but Euclid’s irrational numbers defied representation in the traditional manner.

For Pythagoras, the beauty of mathematics was the idea that rational numbers (whole numbers and fractions) could explain all natural phenomena.

Pythagoras was unwilling to accept that he was wrong, but at the same time he was unable to destroy Hippasus’s argument by the power of logic. To his eternal shame he sentenced Hippasus to death by drowning.

is such a complete body of knowledge that the contents of the Elements would form the geometry syllabus in schools and universities for the next

The mathematician who compiled the equivalent text for number theory was Diophantus

the one detail of Diophantus’s life that has survived is in the form of a riddle said to have been carved on his

His specialty was tackling questions that required whole number solutions, and today such questions are referred to as Diophantine problems.

Of the thirteen books that made up the Arithmetica, only six would survive the turmoils of the Dark Ages

For the next four centuries the Library continued to accumulate books until in A.D. 389 it received the first of two fatal blows, both the result of religious bigotry. The Christian Emperor Theodosius ordered Theophilus, Bishop of Alexandria, to destroy all pagan monuments. Unfortunately when Cleopatra rebuilt and restocked the Library, she decided to house it in the Temple of Serapis, and so the Library became caught up in the destruction of icons and altars. The “pagan” scholars attempted to save six centuries’ worth of knowledge, but before they could do anything they were butchered by the Christian mob. The descent into the Dark Ages had begun.

Then, in 642, a Moslem attack succeeded where the Christians had failed.

is not surprising that most of Diophantus’s work was destroyed; in fact it is a miracle that six volumes of the Arithmetica managed to survive the tragedy of Alexandria.

While Europe had abandoned the noble search for truth, India and Arabia were consolidating the knowledge that had been smuggled out of the embers of Alexandria and were reinterpreting it in a new and more eloquent language.

try multiplying CLV by DCI and you will appreciate the significance of the breakthrough. The equivalent task of multiplying 155 by 601 is a good deal simpler.

the tenth century the French scholar Gerbert of Aurillac learned the new counting system from the Moors of Spain and through his teaching positions at churches and schools throughout Europe he was able to introduce the new system to the West.

Pope Sylvester II,

The vital turning point for Western mathematics did not occur until 1453 when the Turks ransacked Constantinople.

Having survived the onslaught of Caesar, Bishop Theophilus, Caliph Omar, and now the Turks, a few precious volumes of the Arithmetica made their way back to Europe.

Fermat’s judicial responsibilities occupied a great deal of his time, but what little leisure he had was devoted entirely to mathematics. This was partly because judges in seventeenth-century France were discouraged from socializing on the grounds that friends and acquaintances might one day be called before the court.

In effect Diophantus was presenting Fermat with one thousand years’ worth of mathematical understanding.

Friendly numbers are pairs of numbers such that each number is the sum of the divisors of the other number.

Martin Gardner’s book Mathematical Magic Show tells of talismans sold in the Middle Ages that were inscribed with these numbers on the grounds that wearing the charms would promote love.

An Arab numerologist documents the practice of carving 220 on one fruit and 284 on another, and then eating the first one and offering the second one to a lover as a form of mathematical aphrodisiac.

in Genesis Jacob gave 220 goats to Esau. They believed that the number of goats, one half of a friendly pair, w...

No other friendly numbers were identified until 1636, when Fermat discovered the...

Descartes discovered a third pair (9,363,584 and 9,437,056), and Leonhard Euler went on to list sixty-two amicable pairs.

they had all overlooked a much smaller pair of friendly numbers. In 1866 a sixteen-year-old Italian, Nicolò Paganini, discovered the pair 1,184 and 1,210.

During the twentieth century mathematicians have extended the idea further and have searched for so-called “sociable” numbers, three or...

Fermat noticed that 26 is sandwiched between 25 and 27, one of which is a square number (25 = 52 = 5 × 5) and the other is a cube number (27 = 33 = 3 × 3 × 3). He searched for other numbers sandwiched between a square and a cube but failed to find any,

After days of strenuous effort he managed to construct an elaborate argument that proved without any doubt that 26 is indeed the only number between a square and a cube.

While studying Book II of the Arithmetica Fermat came upon a whole series of observations, problems, and solutions that concerned Pythagoras’s theorem and Pythagorean triples. Fermat was struck by the variety and sheer quantity of Pythagorean triples.

Could it really be the case that this minor modification turns Pythagoras’s equation, one with an infinite number of solutions, into an equation with no solutions?

In the margin of his Arithmetica, next to Problem 8, he made a note of his observation:

It is impossible for a cube to be written as a sum of two cubes or a fourth power to be written as the sum of two fourth powers or, in general, for any number which is a power greater than the second to be written as a sum of two like powers.

Among all the possible numbers there seemed to be no reason why at least one set of solutions could not be found,

After the first marginal note outlining the theory, the mischievous genius jotted down an additional comment that would haunt generations of mathematicians:

have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.

This was Fermat at his most infuriating. His own words suggest that he was particularly pleased with this “truly marvelous” proof, but he had no intention of bothering to write out the detail of the argument, never mind publishing it.

Fermat’s notorious discovery happened early in his mathematical career, in around 1637.

On January 9, 1665, he signed his last arrêt, and three days later he died.

Fermat’s discoveries were at risk of being lost forever. Fortunately Fermat’s eldest son, Clément-Samuel, who appreciated the significance of his father’s hobby, was determined that his discoveries should not be lost