Diophantus (ca 250 AD) wrote his famous Arithmetica which was considered by Claude Gaspar Bachet (1581 – 1638) as worthy of deeper study. In 1621 Bachet published his Latin translation of Arithmetica and left wide margins in this publication, which history would later note to be fortunate. In 1637 Fermat, studying the second book of Arithmetica, came across many theorems and observations considering Pythagorean triples. He scibbled a note in the margin of the book, also mentioning an elegant proof for which there was not enough space left.

In 1670, five years after Fermat’s death, his son Samuel found the note of Fermat’s Last Theorem in his copy of Diophantus’s Arithmetica. There Fermat had written

‘I have discovered a truly remarkable proof which this margin is too narrow to contain.’

His famous Theorem states that:

‘Cubem autem in duos cubos, aut quadratoquadtradtum 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 power greater than the second to be written as a sum of two like powers.’

In modern notation, this would mean:

for n > 2  there is no integer solution to the equation    xⁿ + yⁿ = zⁿ.

Historians believe that either there is a remarkably simple proof for this theorem which is yet to be found, or,  far more probable, that Fermat made a mistake in his proof for the general case n > 2  and realized it, since later he only gave the special cases for n = 3 and n = 4 as challenges for other mathematicians. Fermat knew how to prove these special cases but he never mentioned his general proof again.

Fermat himself gave the idea for the special case  n = 4:
 

He proved that there is no right triangle which has area equal a square

there are no    x² + y² = z² such that        ¹/₂.xy = a²         where a an integer.

there are no integers x, y, z with      x⁴ + y⁴ = z⁴

Euler claimed 1753 that he proved the special case for  n = 3:

He showed that for any a, b  the equation
 

p² + 3q² = (a² + 3b²)³ is true if we put

p = a³ – 9ab²    and    q = 3(a²b – b³)            which is fairly straight forward.
 

However his continuation of the proof contained a mistake, namely that numbers of the form  a - b^–3   are of integer structure. Yet this problem can be circumvented using arguments appearing in other proofs of Euler.

It is worth noting that at this stage it remained to prove Fermat’s Last Theorem for odd primes n only. For if there were integers x, y, z

with xⁿ + yⁿ = zⁿ then if n = pq,

                                   (x^q)^p + (y^q)^p = (z^q)^p.

The next major step forward was due to Sophie Germain, who taught herself mathematics and successfully solved problems in number theory. A special case says that if n and 2n+1 are primes then  xⁿ + yⁿ = zⁿ  implies that one of  x, y, z  is divisible by n. Hence Fermat’s Last Theorem splits into two cases.

Case 1: None of x, y, z is divisible by n.

Case 2: One and only one of x, y, z is divisible by n.

Sophie Germain proved Case 1 of Fermat’s Last Theorem for all n < 100:

Her theorem states:
 

If the Fermat equation for exponent p prime has a solution, and if it is a prime with no nonzeor consecutive pth  powers ( mod t ),  then  t must divide one of the numbers  x, y or z.

She used a proof by contradiction in which she tried to prove that x, y and z are non-zero integers satisfying Fermat’s equation where p is prime, and that t is a prime with no nonzero consecutive pth powers mod t, and that t does not divide x, y or z.

For the proof, she used Fermat’s Little Theorem to prove that x has a non-zero inverse (mod t).
Since x, y and z are integer solutions and since t does not divide them, then

   x ^p + y ^p z ^p   mod t    is non-trivial.

Now multiplying this equation by a^p to get   ax ^p + ay ^p az ^p   mod t.

we can reduce it to    1^p + ay ^p az ^p   mod t    because a = x^-1.

so   ay ^p   and  az ^p  must be consecutive non-zero numbers  mod t  since t does not divide a, y or z.

This contradicts the assumption that t has no consecutive non-zero pth  powers.
 

Legendre later extended this method to numbers < 197.
Here it is interesting to note that Germain’s Theorem indicates that counter-examples to Fermat’s Theorem, if any should exist, must be very large. They would have to be numbers with around 30 digits.

At the same time Case 2 still had not been proved for very small n, so it was obvious that this case is where the big problem was.

Later in 1825 Dirichlet and Legendre independently solved half of the case for  n = 5, and in the same year published the proofs.

In 1832 Dirichlet published a proof of Fermat’s Last Theorem for n = 14.

In 1839 Lamé solved the Theorem for the special case   n = 7, in a way which used some new methods.

It was now becoming clear that a general proof was quite improbable, since it seemed that some completely new branches of mathematics had to be established before any attempt could be successful.

In 1847 Lamé announced the final proof of Fermat’s Last Theorem. His proof used factorizing the equation into linear factors over the complex numbers.  But there was an error in his proof which seemed to occur because factorization into primes over complex numbers is not necessarily unique, an argument necessary for Lamé’s proof.

In 1847 Kummer, who proved in one of his works that

if a prime is regular, then Fermat’s Theorem is true for that prime,

managed to prove that

if p does not divide the numerators of any elements of the set of Bernoulli numbers {B₂, B₄, ... B_{p – 3}} then   p is a regular prime.

Here the Bernoulli number  B_n  is defined by the equation:

x/(e^x – 1) =  B_n xⁿ /n!

Note

Bernoulli numbers come from the coefficients in the Taylor expansion of x/(e^x – 1). A recursive definition is possible by defining B_o = 1 and then using

Altenatively, Bernoulli numbers can be defined using Stirling’s formula:

Kummer showed that all primes < 37 are regular.

Later, more complex methods were found by Kummer, Mirimanoff, Furtwaengler, Vandiver and others to establish Fermat’s Theorem for n =37, 59 and 67, the only non-regular primes < 100.

At that time mathematicians generally assumed that the number of regular primes is infinite, but this was also only a conjecture.

In 1915 Jensen proved that the number of irregular primes is infinite

Several institutes and academies offered prizes in the search for the proof of Fermat’s Last Theorem, yet still no proof was found. Over 1000 false proofs were given just between 1908 and 1912.

By 1993, based on Kummer’s Theorem and using large computer calculations, the proof for Fermat’s Last Theorem was made for exponents n < 4 000 000.

In 1983 a big step was taken by Gerd Faltings who proved that

for every n > 2 there are at most a finite number of coprime integers x, y and z with     xⁿ + yⁿ = zⁿ.

In 1955 Taniyama, Weil and Shimura produced a conjecture about elliptic curves, i.e. curves of the form

            y² = x³ + ax + b,   where a, b are constants.

In 1985 the German mathematician Frey conjectured that there was a connection between the Shimura-Taniyama-Weil conjecture and Fermat’s Last Theorem.

Shortly afterwards, in 1986, Ken Ribet managed to prove Frey’s conjecture and show that

a counter-example of Fermat’s Last Theorem would imply a counter-example to the above conjecture.

Finally, the general proof of Fermat’s Last Theorem was found by Andrew Wiles, a mathematics professor at Princeton.

Wiles heard of Ribet’s proof and felt electrified. He set himself the goal of proving the Taniyama-Shimura conjecture. Wiles started in the fall of 1986, secretly, while it was commonly agreed that a proof was not possible in that time. Wiles worked for almost five years alone on the proof using a method called Iwasawa Theory and still could not find an answer.  Then, attending a conference in Boston on elliptic curves, he learned about a new method, the Kolyvagin-Flach method.  In 1993 Wiles decided to confide his friend Nick Katz about his work. Katz was totally amazed. The two of them offered a course on Calculations on Elliptic Curves to test Wiles’ conjectures. After only a few lectures there were only Katz and Wiles left – the material was too difficult.

By May of 1993 Wiles was sure he had solved Fermat’s Last Theorem completely and prepared to represent it at the coming conference in Cambridge in the summer.

In June 1993 Wiles started his 3 lectures about Modular Forms, Elliptic Curves, and Galois Representations. Already during the first lecture his audience realized that he was going in a certain direction. Rumors started. In the second lecture he had an essentially larger audience, and he showed clearly that he was going to prove the Taniyama-Shimura conjecture. In the thrid lecture most of the mathematicians who contributed to the proof were there. Everyone was just waiting for the grand finale. Tension made it almost impossible for Wiles’' colleagues to wait to see his final move. After Wiles proved the conjecture, he said to his audience :

I think I should stop here.

The audience, hearing these words, erupted into long applause, most of them feeling that they had had the opportunity to attend a historical event.

But the story did not end with the unbelievably large and incredibly complex proof. In August 1993 Katz discovered a slight error in the proof, which Wiles could not repair. In December 1993 Wiles admitted that there was a gap in his proof which he could not fill and that, strictly speaking, his ‘proof’ was not yet a proof.

Weil wrote in Scientific American:

... To some extent, proving Fermat’s Theorem is like climbing Everest. If a man wants to climb Everest and falls short of it by 100 yards, he has not climbed Everest.

Taylor suggested an attempt to extend Flach’s method and Wiles agreed on it. After two weeks of work, inspiration suddenly came to Wiles.

“I was sitting at my desk examining my method. It wasn’t that I thought I could make it work, but I thought I could at least explain why it didn’t work. Suddenly, I had this incredible revelation. I realized although Kolyvagin-Flach wasn’t working completely, it was all I needed to make my original Iwasawa theory work! [...] It was so indescribably beautiful; it was so simple and so elegant. I couldn’t understand how I’d missed it and I just stared at it in disbelief for twenty minutes. During the day I walked around, and I’d keep coming back to see if it was still there. It was. [...] Nothing I ever do again will mean as much.”

On October 1994 Wiles sent his corrected proof to colleagues including Faltings. The new proof was significantly shorter and simpler.

Now we believe that Wiles has truly proved Fermat’s Theorem, even though there are not so many people in the world who can understand it completely. Maybe a handful. But even if a small doubt should still stay, almost all the leading mathematicians agree that Wiles’ proof is correct.

The mathematical Mount Everest was climbed by Andrew Wiles in 1994, after almost 360 years of search, hope and fails.

Here, a truly remarkable piece of mathematical adventure finally found its happy end.

Here is the roughest outline of the proof, for interested, but not skilled amateurs:

Basically, the proof consists of two main theorems.

Theorem 1

If there is a solution to Fermat's equation, then the elliptic curve defined by the equation

is semistable, but not modular.

Theorem 2

All semistable elliptic curves with rational coefficients are modular.

The beauty of these theorems lies in the connection between elliptic curves and the modular forms represented by them.  Theorem 2 in particular states that the parallels, which were observed in the theory of each, do actually result from a fundamental connection between the two mathematical objects.

For the proof for Fermat’s Last Theorem it is sufficient to show the theorems with the restriction of the curves being semistable, but many mathematicians believe that greater generalizations should still be possible. This is one of the most modern and promising research area in mathematics today.

Andrew Wiles    1953 –

Pierre de Fermat    1601 – 1665