By Griet Christiaens

               The  Prince  of  Amateurs  of  Mathematics 

The Background of Pierre de Fermat

Fermat was baptized on August 20, 1601 (this is accepted as his date of birth in both mathematical and historical communities) in the town of Beaumont-de-Lomagne, France, near Montauban. He was born into a Catholic family as one of four children; three boys and one girl. His father was a prosperous leather merchant and the second consul of his hometown, and his mother, a parliamentary noblesse de la robe. It is easy to see that they were well-to-do. His uncle, who was his godfather, was also a merchant.

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He began his secondary schooling at the convent of Cordeliers, an institution run by Franciscans in France. After that it is believed that he may have attended the University of Toulouse.

He acquired his degree of Bachelor of Civil Laws from the University of Orleans in 1631.
Fermat was financially secure all throughout his life. He registered as a lawyer in Toulouse in 1631. In the same year he purchased his way into Parliament for 43,500 livres. When he married, his wife brought a dowry of 12, 000 livres. For his position in congress he earned the ‘de’ which is in his name.

Frequently Fermat socialized with Carcavi, Mersenne, Blaise Pascal,Wallis, Digby and many other great discoverers of his time. He communicated with all the great mathematicians in Europe, posing and solving problems, disseminating ideas of various math subjects and inventing new mathematics such as probability theory.  His Unique Integral Solution was actually issued as a challenge to the English mathematicians Wallis and Digby. Fermat used Francois Viete’s algebra to restore Apollonius of Perga’s Plane loci. The restored work was renamed Method for determining Maxima and Minima and Tangents to Curved Lines. Decartes and Fermat, indepent from each other, invented analytic geometry.
Fermat also conceived the idea of differential calculus although he never reduced it to a set rule of thumb.

Pierre de Fermat died in Castres, France (somewhere near Toulouse) on the 12th of January, 1665 after a lifetime of 64 years. Although Fermat discovered many different things in many different fields in his lifetime, he published very little besides a few papers. He communicated most of his results in letters to friends (usually without proof). Most of his discoveries were written in margins, therefore it is very difficult to date each of his findings. 

Fermat also worked on optics and his "principle of least time" was generalized in the nineteenth century to enable the behaviour of a variety of physical systems to be interpreted. But he is widely known for his contribution to whole number theory and certainly for his "last theorem". He dabbled often in the Unique Integer Solutionfield of ‘Number Theory’. One of these ‘dabblings’ is the Unique Integral Solution. With this equation Fermat was following Diophantus, a Greek mathematician who prospered in the 3rd century AD, also known as ‘The Father of Algebra.’, and his Diophantine Analysis. He was a lawyer by profession but he occupied himself in the fields of number theory, calculus, analytical geometry and probability just as a recreational hobby !

His favourite recreation : exploration of the properties of numbers

The theory of numbers appears to have been the favourite study of Fermat. He prepared an edition of Diophantus, and the notes and comments thereon contain numerous theorems of considerable elegance. Most of the proofs of Fermat are lost, and it is possible that some of them were not rigorous - an induction by analogy and the intuition of genius sufficing to lead him to correct results. The following examples will illustrate these investigations.

         a) If p is a prime and is no divisor of  a then a p - 1 - 1 is divisible by p, that is equal to p (mod p). The first proof given by Euler in 1736 of this so called
         "small theorem of Fermat" is well known.             

b) An odd prime can be expressed as the difference of two square integers in one and only one way. Fermat’s proof is as follows. Let n be the prime, and suppose it equals to x² - y², that is, to (x + y)(x - y). Now, by hypothesis, the only integral factors of n are n and unity, hence x + y = n and x - y = 1. Solving these equations we get x = ½ (n + 1) and y = ½ (n - 1).

c) He gave a proof of the statement made by Diophantus that the sum of the squares of two integers cannot be of the form 4n - 1; and he added a corollary probably meaning that it is impossible that the product of a square and a prime of the form 4n - 1 [even if multiplied by a number prime to the latter], can be either a square or the sum of two squares.

For example, 44 is a multiple of 11 (which is of the form 4 × 3 - 1) by 4, hence it cannot be expressed as the sum of two squares.

He also stated that a number of the form a² + b², where a is prime to b, cannot be divided by a prime of the form 4n - 1.

d) Every prime of the form 4n + 1 is expressible, and that in one way only, as the sum of two squares. This problem was first solved by Euler, who showed that a number of the form (4n + 1) can be always expressed as the sum of two squares.

e) No integral values of x, y, z can be found to satisfy the equation  xn + yn   =  zn ; if n be an integer greater than 2. This proposition has acquired extraordinary celebrity from the fact that no general demonstration of it has been given, but there is no reason to doubt that it is true.

Probably Fermat discovered its truth first for the case n = 3, and then for the case n = 4.

His proof for the former of these cases is lost, but that for the latter is extant, and a similar proof for the case of n = 3 was given by Euler. These proofs depend on showing that, if three integral values of x, y, z can be found which satisfy the equation, then it will be possible to find three other and smaller integers which also satisfy it: in this way, finally, we show that the equation must be satisfied by three values which obviously do not satisfy it. Thus no integral solution is possible. It would seem that this method is inapplicable to any cases except those of n = 3 and n = 4.

Last theorem (supposition?) of Fermat

By far the most famous is the one called Fermat’s Last Theorem:

"It is impossible to write a cube as a sum of two cubes, a fourth power as a sum of fourth powers, and, in general, any power beyond the second as a sum of two similar powers.  For this, I have found a truly wonderful proof, but the margin is too small to contain it." [This was written by Fermat near 1637, in Latin.]

Now we would write this as follows. (Wiles 1995)

The equation xn + yn = zn has no solutions in positive integers for n greater than 2. So for n > 2 the only solution is (0,0,0).

Few (now) believe Fermat had found the proof he claimed. Because this lost? proof has been the source of mystery and discussion in the mathematical world for nearly four centuries.

In 1908, Paul Wolfshehl, a little-known mathematics professor in Darmstadt, Germany, left a reward in his will of 100,000 DM to the mathematician who could prove Fermat's last theorem. In 1992 J.P. Buhler and R. Crandall used super-computer technology to verify Fermat's last theorem for exponents as large as 4,000,000 but this does not constitute a proof !

wiles9.jpg (2779 bytes) Princeton mathematician Prof. Andrew Wiles, graduated from Oxford and holding a Ph.D. from Cambridge, found the first accepted proof after 3 years of search. This was in 1995, some 350 years later. 

Wiles' proof is exceptionally long, more than 200 pages, and difficult... no more than half a dozen people in the world are capable of fully understanding  it !

 

Sources

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