The Books: “Fermat’s Enigma: The Epic Quest to Solive the World’s Greatest Mathematical Problem” (Simon Singh)

Next book on the science and philosophy shelf:

FermatsEnigma.jpgA book about 17th century French mathematician Pierre de Fermat and his last theorem. Proving this last theorem turned out to be no easy feat, and mathematicians tried, for 350 years. It has been called “the Holy Grail of mathematics”. Obviously, I’m pulling this book down from my “Math and Science for People who Love Math and Science but Don’t Understand the Actual Math and Science” shelf. One of my favorite shelves! The book is by Simon Singh and it’s called Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem.

The book mostly details the mathematicians throughout history who have struggled to find a proof for Fermat’s Last Theorem. But the following excerpt is about Fermat, and his “enigma”:


EXCERPT FROM Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem:

One of Fermat’s discoveries concerned the so-called friendly numbers, or amicable numbers, closely related to the perfect numbers that had fascinated Pythagoras two thousand years earlier. Friendly numbers are pairs of numbers such that each number is the sum of the divisors of the other number. The Pythagoreans made the extraordinary discovery that 220 and 284 are friendly numbers. The divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, and the sum of all these is 284. On the other hand, the divisors of 284 are 1, 2, 4, 71, 142, and the sum of all these is 220.

The pair 220 and 284 was said to be symbolic of friendship. 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. Early theologians noted that in Genesis Jacob gave 220 goats to Esau. They believed that the number of goats, one half of a friendly pair, was an expression of Jacob’s love for Esau.

No other friendly numbers were identified until 1636, when Fermat discovered the pair 17,296 and 18,416. Although not a profound discovery, it demonstrates Fermat’s familiarity with numbers and his love of playing with them. Fermat started a fad for finding friendly numbers; Descartes discovered a thir pair (9,363,584 and 9,437,056), and Leonhard Euler went on to list sixty-two amicable pairs. Curiously they had all overlooked a much smaller pair of friendly numbers. In 1866 a sixteen-year-old Italian, Nicolo 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 more numbers that form a closed loop. For example, in this loop of five numbers (12,496; 14,288; 15,472; 14,536; 14,264) the divisors of the first number add up to the second, the divisors of the second add up to the third, the divisors of the third add up to the fourth, the divisors of the fourth add up to the fifth, and the divisors of the fifth add up to the first. [Note from Sheila: Cool!!!]

Although discovering a new pair of friendly numbers made Fermat something of a celebrity, his reputation was truly confirmed thanks to a series of mathematical challenges. For example, Fermat noticed that 26 is sandwiched between 25 and 17, one of which is a square number (25 = 52 = 5 x 5) and the other is a cube number (27 = 33 = 3 x 3 x 3). He searcherd for other numbers sandwiched between a square and a cube but failed to find any, and suspected that 26 might be unique. 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. His step-by-step logical proof established that no other numbers could fulfill this criterion.

Fermat announced this unique property of 26 to the mathematical community, and then challenged them to prove that this was the case. He openly admitted that he himself had a proof; the question was, however, did others have the ingenuity to match it? Despite the simplicity of the claim the proof is fiendishly complicated, and Fermat took particular delight in taunting the English mathematicians Wallis and Digby, who eventually had to admit defeat. UYltimately Fermat’s greatest claim to fame would turn out to be another challenge to the rest of the world. However, it would be an accidental riddle that was never intended for public discussion.

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. He was aware that centuries earlier Euclid had stated a proof which demonstrated that, in fact, there are an infinite number of Pythagorean triples. Fermat must have gazed at Diophantus’s detailed exposition of Pythagorean triples and wondered what there was to add to the subject. As he stared at the page he began to play with Pythagoras’s equation, trying to discovere something that had evaded the Greeks.

Suddenly, in a moment of genius that would immortalize the Prince of Amateurs, he created an equation that, though very similar to Pythagoras’s equation, had no solutions at all…

Instead of considering the equation

x2 + y2 = z2,

Fermat was contemplating a variant of Pythagoras’s creation:

x3 + y3 = z3.

As mentioned in the last chapter, Fermat had merely changed the power from 2 to 3, the square to a cube, but his new equation apparently had no whole number solutions whatsoever. Trial and error soon shows the difficulty of finding two cubed numbers that add together to make another cubed number. 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?

He altered the equation further by changing the power to numbers bigger than 3, and discovered that finding a solution to each of these equations was equally difficult. According to Fermat there appeared to be no three numbers that would perfectly fit the equation

xn + yn = zn where n represents 3,4,5…

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

Cubem autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere.

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 greater power 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, yet Fermat stated that nowhere in the infinite universe of numbers was there a “Fermatean triple”. It was an extraordinary claim, but one that Fermat believed he could prove. After the first marginal note outlining the theory, the mischievous genius jotted down an additional comment that would haunt generations of mathematicians:

Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet.

I 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. He never told anyone about his proof, and yet, despite the combination of indolence and modesty, Fermat’s Last Theorem, as it would later be called, would become famous around the world for centuries to come.

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4 Responses to The Books: “Fermat’s Enigma: The Epic Quest to Solive the World’s Greatest Mathematical Problem” (Simon Singh)

  1. Ken Summers says:

    Small disappointment: I had hoped that the Nicolo Paganini who discovered that pair of friendly was Nicolo Paganini the violinist. Sadly, he wasn’t.

  2. Bernard says:

    You ARE going to tell us how this “truly marvelous” proof was finally verified?

  3. red says:

    Bernard – you have to read the book to find out!! It’s a gripping mystery. :)

  4. Ken Summers says:

    Hate to break it to you, Bernard, but I know how it ends…

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