Fermat's Enigma

by Simon Singh

if it were not for Clément-Samuel, the enigma known as Fermat’s Last Theorem would have died with its creator.

Clément-Samuel spent five years collecting his father’s notes and letters, and examining the jottings in the margins of his copy of the Arithmetica.

Alongside Bachet’s original Greek and Latin translations were forty-eight observations made by Fermat.

Once Fermat’s Observations reached the wider community, it was clear that the letters he had sent to colleagues represented mere morsels from a treasure trove of discovery. His personal notes contained a whole series of theorems.

Leonhard Euler, one of the greatest mathematicians of the eighteenth century, attempted to prove one of Fermat’s most elegant observations, a theorem concerning prime numbers.

All prime numbers (except 2) can be put into two categories: those that equal 4n + 1 and those that equal 4n – 1, where n equals some number.

Fermat’s prime theorem claimed that the first type of primes were always the sum of two squares (13 = 22 + 32

the second type could never be writte...

in 1749, after seven years’ work and almost a century after Fermat’s death, Euler succeeded in proving this prime number theorem.

Fermat’s panoply of theorems ranged from the fundamental to the simply amusing.

Unsubstantiated ideas are infinitely less valuable and are referred to as conjectures.

it is called the “Last” Theorem because it remains the last one of the observations to be proved.

Leonhard Euler was born in Basle in 1707, the son of a Calvinist pastor, Paul Euler.

Leonhard dutifully obeyed and studied theology and Hebrew at the University of Basle.

Fortunately for Euler the town of Basle was also home to the eminent Bernoulli clan.

Bernoulli family was to mathematics what the Bach family was to music.

Daniel and Nikolaus Bernoulli were close friends of Leonhard Euler,

During the era of Fermat, mathematicians were considered amateur number-jugglers, but by the eighteenth century they were treated as professional problem-solvers.

was partly a consequence of Sir Isaac Newton and his scientific calculations.

and even when he was cradling an infant in one hand Euler would be outlining a proof with the other.

for the specific case n = 4 elsewhere in his copy of the Arithmetica and incorporated it into the proof of a totally different problem. Even though this is the most complete calculation he ever committed to paper, the details are still sketchy and vague,

Despite the lack of detail in Fermat’s scribbles, they clearly illustrate a particular form of proof by contradiction known as the method of infinite descent.

On August 4, 1753, Euler announced in a letter to the Prussian mathematician Christian Goldbach that he had adapted Fermat’s method of infinite descent and successfully proved the case for n = 3. After a hundred years this was the first time anybody had succeeded in making any progress toward meeting Fermat’s challenge.

to cover the case n = 3 Euler had to incorporate the bizarre concept of a so-called imaginary number, an entity that had been discovered by European mathematicians in the sixteenth century.

we forget that there was a time when some of these numbers were not known. Negative numbers, fractions, and irrational numbers all had to be discovered, and the motivation in each case was to answer otherwise unanswerable questions.

It is unthinkable for mathematicians not, in theory at least, to be able to answer every single question, and this necessity is called completeness.

Mathematicians express this by saying that fractions are necessary for completeness.

What is the square root of two? The demand for completeness meant that yet another colony was added to the empire of numbers.

The Italian mathematician Rafaello Bombelli was studying the square roots of various numbers when he stumbled upon an unanswerable question. The problem began by asking, What is the square root of one, ?

The solution for Bombelli was to create a new number, i, called an imaginary number, which was simply defined as the solution to the question, What is the square root of negative one?

The real story wazs a bit different

It is easier to accept the concept of –1 only because we have experience of the analogous concept of “debt,” whereas we have nothing in the real world to underpin the concept of an imaginary number.

Mathematicians resolve this crisis by creating a separate imaginary number line that is perpendicular to the real one, and that crosses at zero,

combinations of real and imaginary numbers (e.g., 1+2i), called complex numbers, live on the so-called number

Euler showed that by incorporating the imaginary number, i, into his proof he could plug holes in the proof and force the method of infinite descent to work for the case n = 3.

he could not repeat for other cases of Fermat’s Last Theorem.

His loss of sight began in 1735 when the Academy in Paris offered a prize for the solution to an astronomical problem.

He became obsessed with the task, worked continually for three days, and duly won the prize. However, poor working conditions combined with intense stress cost Euler, then still only in his twenties, the sight of one eye.

Euler was replaced by Joseph-Louis Lagrange as mathematician to the court of Frederick the Great,

Euler claimed that “now I will have less distraction.”

began to practice writing with his fading eye closed in order to perfect his technique before the onset of darkness. Within weeks he was blind.

Euler continued to produce mathematics for the next seventeen years,

Colleagues suggested that the onset of blindness appeared to expand the horizons of his imagination.

lunar positions were completed during his period of blindness.

In 1776 an operation was performed

few days Euler’s sight seemed to have been restored. Then infection set in

he continued to work until, on September 18, 1783, he suffe...

Marquis de Condorcet, “Euler ceased to live ...

Fermat had given mathematicians a head start by providing them with the proof that there were no solutions to the equation

Euler had adapted the proof to show that there were no solutions to

The proof for the case n = 4 also proves the cases n = 8, 12, 16, 20,….The reason is that any number that can be written as an 8th (or a 12th, 16th, 20th,…) power can also be rewritten as a 4th power.