The Mathematics of Pierre de Fermat (1601 - 1665) (French)

Pierre de Fermat     Pierre de Fermat is perhaps the most well-known mathematician in the world. This is due to a conjecture stated by him, the famous (or infamous!) Fermat's Last Theorem (FLT). It states that there are no integral values of x, y, z which can satisfy the equation xn + yn = zn, where n is an integer>2.

This problem has remain unsolved for 3 centuries (until recently). There are many other unsolved problems in Mathematics, so why is FLT such a big deal? The reason is that Fermat mentioned he found the proof, but that the margin in the book was too narrow to contain the proof. His proof was never found (perhaps it did not exist at all, Fermat might have made a mistake in his 'proof') and nobody until recently could prove it either. Not even the great Prince of Mathematics, Gauss, could settle the issue. Of course, FLT is now fully resolved and we owe it to Andrew Wiles.

Fermat was educated at home. As such, we can say he is an amateur mathematician. Despite this, he has contributed much to Mathematics, especially to the field of Number Theory. Except a few isolated papers, Fermat published nothing in his lifetime, and gave no systematic exposition of his methods. Some of the most striking of his results were found after his death on loose sheets of paper or written in the margins of works which he had read and annotated, and are unaccompanied by any proof.

1. Number Theory

Fermat's greatest contribution to Mathematics is in Number Theory. He solved quite a number of Diophantine equations, i.e. equations with integral solutions. One of these is Fermat's Little Theorem:

If p is a prime and a is relatively prime to p, then ap-1 - 1 is divisible by p. In congruence notation, ap-1 - 1 ≡ 0 (mod p).

This was first proved by Leonhard Euler.

2. Infinite Descent

Basically, this method of proof aims to show that

"If you can find a solution to a problem, then you can find another solution with smaller values indefinitely".

He used this method to show that the area of a Pythagorean triangle cannot be a square:

"If the area of such a triangle were a square, then there would also be a smaller one with the same property, and so on, which is impossible".

3. Probability Theory

Fermat shares the honour of being the discoverer of Probability theory with Blaise Pascal. His involvement in the founding of probability came from a series of questions posed by Pascal to him. One of these is: Consider two players, A and B. A needs 2 points to win and B needs 3 points. Then the game will be certainly decided in the course of 4 trials. Now there are 16 combinations, namely, 
     aaaa   aaab   aaba   aabb
     abaa   abab   abba   abbb
     baaa   baab   baba   babb
     bbaa   bbab   bbba   bbbb
Now, every combination in which a occurs 2 or more times represents a win for A, and every combination in which b occurs 3 or more times represents a win for B. Thus, on counting them, it will be found that there are 11 cases in which A wins, and 5 cases in which B wins. If these cases are all equally likely, A's chance of winning the game is 11/16 and B's chance is 5/16. Click here for a generalization of the problem.

4. Fermat Numbers

Fermat Numbers are integers of the form:
Fn = 22n + 1, where n is a non-negative integer.

Since F0=3, F1=5, F2=17, F3=257, and F4=65537 Fermat induced that all of these numbers are primes, but later Euler proved ingeniously that 641 divides F5. In fact, for n between 4 and 31, Fn is a composite. A quick means of testing the primality of Fn is via Pepin's Test:

Fn is prime if and only if 3(Fn-1)/2 = -1 (mod Fn).

5. Fermat Polygonal Number Theorem

He proposed that every positive integer is a sum of at most 3 Triangular Numbers, 4 Square Numbers, 5 Pentagonal Numbers, and n N-gonal Numbers. Fermat claimed to have a proof of this result (by infinite descent), but like most of his proofs, it has never been found. Euler proved an identity, Euler identity, which was subsequently used by Lagrange to prove the square case in 1772. Jacobi independently proved it also. Gauss proved the triangular case in 1796. Finally, in 1813, Cauchy proved the general case.

Short notes on some of the mathematicians mentioned in this page.
1) Joseph Louis Lagrange (1736-1813) (French) contributed to many fields of Mathematics, a true universalist.

2) Jacobi ()

3) Augustin-Louis Cauchy (1789-1857) (French) contributed greatly to Complex Analysis on top of Differential Equations and Group Theory.

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