Fermat's Enigma

by Simon Singh

Euler’s proof for the case n = 3 automatically proves the cases n = 6, 9, 12, 15,…

The proof for the case n = 3 is particularly significant because the number 3 is an example of a prime number.

Number theorists consider prime numbers to be the most important numbers of all, because they are the atoms of mathematics.

To prove Fermat’s Last Theorem for all values of n, one merely has to prove it for the prime values of

But how can something that is undeniably smaller than an infinite quantity also be infinite?

David Hilbert said: “The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite.”

Hilbert’s Hotel

in fact it can be proved that the infinite set of irrational numbers is larger than the infinite set of rational numbers.

In the 1970s Whitfield Diffie and Martin Hellman came up with the idea of looking for a mathematical process that was easy to perform in one direction but incredibly difficult to perform in the opposite direction.

In 1977 Ronald Rivest, Adi Shamir, and Leonard Adleman, a team of mathematicians and computer scientists at the Massachusetts Institute of Technology, realized that prime numbers were the ideal basis for an easy-scramble/hard-unscramble process.

Sophie Germain lived in an era of chauvinism and prejudice, and in order to conduct her research she was forced to assume a false identity, study in terrible conditions, and work in intellectual isolation.

The first woman known to have made an impact on the subject was Theano in the sixth century B.C., who began as one of Pythagoras’s students before becoming one of his foremost disciples and eventually marrying him.

Pythagoras is known as the “feminist philosopher” because he actively encouraged women scholars, Theano being just one of the twenty-eight sisters in the Pythagorean Brotherhood.

In later centuries the likes of Socrates and Plato would continue to invite women into their schools,

Hypatia, the daughter of a mathematics teacher at the University of Alexandria, was famous for giving the most popular discourses in the known world

when asked why she never married she replied that she was wedded to the truth.

On a fatal day, in the holy season of Lent, Hypatia was torn from her chariot, stripped naked, dragged to the church, and inhumanely butchered by the hands of Peter the Reader and a troop of savage and merciless fanatics; her flesh was scraped from her bones with sharp oyster-shells, and her quivering limbs were delivered to the flames.

Maria Agnesi was born in Milan in 1718 and, like Hypatia, was the daughter of a mathematician.

particularly famous for her treatises on the tangents to curves.

curves were called versiera, a word derived from the Latin vertere, “to turn,” but it was also an abbreviation for av...

was mistranslated into English as the “witch of Agnesi,” and in time the mathematician herself was r...

Institutionalized discrimination against women continued right through to the twentieth century, when Emmy Noether, described by Einstein as “the most significant creative mathematical genius thus far produced since the higher education of women began,” was denied a lectureship at the University of Göttingen.

Later her colleague Edmund Landau was asked whether Noether was indeed a great woman mathematician, to which he replied: “I can testify that she is a great mathematician, but that she is a woman, I cannot swear.”

Of all the European countries France displayed the most chauvinistic attitude toward educated women,

Sophie Germain revolutionized the study of Fermat’s Last Theorem and made a contribution greater than any of the men who had gone before her.

Sophie Germain was born on April 1, 1776, the daughter of a merchant, Ambroise-François Germain.

she resorted to covertly studying at the Ecole by assuming the identity of a former student at the academy, Monsieur Antoine-August Le Blanc.

Everything was going well until a couple of months later when the supervisor of the course, Joseph-Louis Lagrange, could no longer ignore the brilliance of Monsieur Le Blanc’s answer sheets.

Lagrange, who was one of the finest mathematicians of the nineteenth century, requested a meeting with the reformed student and Germain was forced to reveal her true identity. Lagrange was astonished and pleased to meet the young woman

Gauss is acknowledged as being one of the most brilliant mathematicians who has ever lived.

Germain had first encountered his work through studying his masterpiece Disquisitiones arithmeticae,

Gauss replied, “I am very much obliged for your news concerning the Paris prize. But I confess that Fermat’s Last Theorem as an isolated proposition has very little interest for me, for I could easily lay down a multitude of such propositions, which one could neither prove nor disprove.”

Seventy-five years earlier Euler had published his proof for the case n =

her immediate goal was not to prove one particular case, but to say something about many cases

In her letter to Gauss she outlined a calculation that focused on a particular type of prime number p such that (2p + 1) is also prime.

Legendre was a man in his seventies who had lived through the political turmoil of the French Revolution.

Dirichlet was an ambitious young number theorist who had only just turned twenty.

Fourteen years later the French made another breakthrough. Gabriel Lamé made some further ingenious additions to Germain’s method and proved the case for the prime n = 7. Germain had shown numbers theorists how to destroy an entire section of prime cases, and now it was up to the combined efforts of her colleagues to continue proving Fermat’s Last Theorem one case at a time.

When Germain wrote to Gauss she was still in her twenties,

she feared that the great man would not take her seriously

Germain resorted once again to her pseudonym, signing her letters ...

Gauss, unaware of his correspondent’s true identity, attempted to put Germain at ease and replied: “I am delighted that arithmetic has found in you so able a friend.”

In 1806 Napoleon was invading Prussia and the French army was storming through one German city after another.

might also take the life of her other great hero, Gauss, so she sent a message to her friend General Joseph-Marie Pernety, who was in charge of the advancing forces. She asked him to guarantee Gauss’s safety,

In Germain’s next letter to Gauss she reluctantly revealed her true identity.

Gauss wrote back to her with delight:

that the attractions of this science, which has enriched my life with so many joys, are not chimerical, as the predilection with which you have honored it.

in 1808 the relationship ended abruptly. Gauss had been appointed professor of astronomy at the University of Göttingen, his interest shifted from number theory to more applied mathematics, and he no longer bothered to return Germain’s letters. Without her mentor her confidence began to wane, and within a year she abandoned pure mathematics.

As a result of this research and her work on Fermat’s Last Theorem she received a medal from the Institut de France, and became the first woman who was not a wife of a member to attend lectures at the Academy of Sciences.

When the Eiffel Tower was erected, in which the engineers were obliged to give special attention to the elasticity of the materials used, there were inscribed on this lofty structure the names of seventy-two savants. But one will not find in this list the name of that daughter of genius, whose researches contributed so much towards establishing the theory of the elasticity of metals—Sophie Germain.