Yogita Chellani

Term Paper, History of Mathematics, Rutgers

The French mathematician Pierre de Fermat(1601-1665) was possibly the most productive mathematician of his era, making many contributions, some of which were to calculus, number theory, and the law of refraction. We will survey those contributions here, paying particular attention to his work in number theory.

The following account of Fermat's background is taken from Mahoney's book, The Mathematical Career of Pierre de Fermat. Pierre de Fermat was born on August 17, 1960, in Beaumont-de-Lomagne, a small town near Toulouse in the south part of France, near the border with Spain. His father, Dominique Fermat, was a wealthy leather merchant who held the position of "second consul" of Beaumont-de-Lomagne, a governmental position similar to the position of mayor in our time. His mother, Claire, née de Long, was the daughter of a prominent family. Fermat had a brother, Clément, and two sisters, Louise and Marie.

While relatively little is known of Fermat's early education, it is known that he was of Basque origin and received his primary and secondary education at the monastery of Grandsl ve, run by the Cordeliers (Franciscans), in Beaumont-de-Lomagne. For his advanced studies he first attended the University of Toulouse before moving to Bordeaux in the second half of the 1620's. In Bordeaux (1629) Fermat began his first serious mathematical researches, where he gave a copy of his restoration of Appollonius's Plane Loci to one of the mathematicians there. In Bordeaux he contacted Beaugrand and during this time he produced work on maxima and minima. He gave his work to Etienne de'Espagnet, who shared mathematical interests with Fermat.

From Bordeaux Fermat went to study at the University of Law at Orléans. On May 1, 1631 he received the degree of Bachelor of Civil Laws. Fermat's choice of a legal career was natural and typical of his time, for his father's wealth and his mother's famil y background. To be in this career was an avenue to a higher social status and political power. After graduating he purchased the office of councillor at the parliament in Toulouse. Soon after that he acquired a wife. She was his cousin fourth removed, Lo uise de Long. He gave a dowry of 12,000 livres, which was not a problem for the young lawyer. Soon after, he served in the local parliament and became councillor, or legislator. His entire family, now including his father-in-law, were members of the upper class. Mahoney tells us how this affected his social status as well.

The "de" is the mark of nobility in France.

Very little is known about Fermat's private life. He had five children, Clément-Samuel, Jean, Claire, Catherine, and Louise. Clément-Samuel was the oldest and closest to Fermat. He may have shared many mathematical interests with Fermat. Clément-Samuel eventually inherited his father's office of councillor.

For the remainder of his life Fermat lived in Toulouse, but he also worked in his hometown of Beaumont-de-Lomagne, and the nearby town of Castres. First he worked in the lower chamber of Parliament, but then in 1638 he was appointed to the higher chamber , and finally in 1652 he was promoted to the highest level in the criminal court. This position was usually given to people of seniority, but since the plague had struck in the early 1650's, many of the older men had died. Fermat himself was struck down w ith the plague. In 1653 his death was wrongly reported; Fermat had survived. This account of Fermat's background and life was taken from [Mahoney, pp. 15-17].

We now call this process integral calculus. It is ironic that Fermat did not see what we now call the Fundamental Theorem of Calculus. However, his work on this subject was an aid to developing the differential calculus. In addition to his contribution t o calculus, Fermat contributed to the law of refraction. He had a disagreement with the philosopher and amateur mathematician, René Descartes. "According to [Fermat's] principle, if a ray of light passes from a point A to another point B, being reflected and refracted(refracted; that is, bent, as in passing from air to water, or through a jelly of variable density) in any manner during the passa ge, the path which it must take can be calculated- all its twistings and turnings due to the refraction, and all its dodgings back and forth due to reflections- from the single requirement that the time taken to pass from A to B shall be an extreme." (Bel l, p.63).

Descartes was trying to justify the sine law (Snell's law) by saying that light travels more rapidly in the denser of the two media involved in the refraction. Twenty years later Fermat realized that this appeared to be in conflict with the Aristotelian view that nature always chooses the shortest path.

Fermat applied his method of maxima and minima and made the assumption that light travels less rapidly in the medium. He showed that the law of refraction is consistent with the principle of least time. "From this principle Fermat deduced the familiar laws of reflection and refraction: the angle of reflection; the sine of the angle of incidence(in refraction)is a constant number times the sine of the angle of refraction in passing from one medium to anot her." (Bell, p.63).

Finally I will discuss Fermat's contribution to number theory in greater detail than his other contributions, as Bell does: "We begin with a famous statement Fermat made about prime numbers." Bell first reminds us that: "A positive prime number, or briefly a prime number is any number greater than 1 which has as its divisor(without remainder) only 1 and the number itself; for example 2,3,5,7,13,17, are primes and so are 257,65537" (Bell, p.65)

He then explains that Fermat observed that the numbers 3,5,17,257,65537, all belong to one sequence and can be generated by one simple process: 3=2+1, 5=2^2+1, 17=2^4+1, 257=2^8+1, 65537=2^(16)+1. So thus the sequence seen here is 2^(2^n)+1, for n=1 to infinity. Now if we wanted to check if one of these numbers is prime it would not be an easy process, unless you follow Fermat, who was claiming that all numbers of the previous sequence are prime. But Fermat was mistaken; he just guessed and did not prove his idea. Euler, a century later, showed that 232 +1 has 641 as a factor. So Fermat was wrong and we still do not know whether there are any primes among the Fermat numbers, for n>4.

Although Fermat made a mistake, through his work on numbers he discovered that every prime number of the form 4n+1 is expressible as the sum of two squares. However, Fermat as in almost all his mathematical work left no written proof of this theorem. He did however write a letter to his friend and mathematician, Carcavi. In this letter he included how he proved this theorem. Bell tells us that in his letter he wrote, "The course of my reasoning in affirmative propositions is such: if an arbitrarily chosen prime of the form 4n+1 is not a sum of two squares, [I prove that] there will be another of the same nature, less than the one chosen, and [therefore] next a third s till less and so on." (Bell, p.70).

So Fermat comes to the number 5 which is the least of all these numbers. He sees that then 5 is not a sum of two squares. However, it is. Then Fermat says in the letter according to Bell, "Therefore we must infer by a reductio ad absurdum that all numbers of the form 4n+1 are sums of two squares." (Bell, p.70).

This causes confusion. Fermat is not explicit as to how he proved his statement. We see that Fermat is using a device that he called the method of "infinite descent". This is known as an inverted form of reasoning by recurrence or mathematical induction. A more important result derived from this is what is now known as Fermat's Lesser Theorem. The theorem is that if p is a prime number and is a is any positive integer, then ap - a is divisible by p. Once again no proof was given. In this case Gotfried Lei bniz, the 17th Century German mathematician and philosopher, and Leonhard Euler, the 18th Century Swiss mathematician, provided proofs.

Finally, I will discuss briefly Fermat's last discovery, Fermat's Last Theorem. While he was reading a copy of Diophantus' Arithmetica, Fermat made a marginal note that remained unsolved until very recently. He read the eighth problem in Dipohantus's boo k, which asks for the solution in rational numbers of the equation x2 + y2= a2. According to Bell Fermat says, "On the contrary, it is impossible to separate a cube into two cubes, a fourth power into two fourth powers, or generally, any power above the second into two powers of the same degree: I have discovered a truly marvelous demonstration [of this general theorem] which this margin is too narrow to contain" (Bell, p.71).

To restate this, Fermat is saying that no nonzero whole numbers or fractions exists with xn + yn = an, if n is a whole number greater than 2. However, Fermat did not live to prove what he had in mind. This theorem has puzzled mathematicians for many years until recently, and is still remembered as Fermat's Last Theorem.

In conclusion, Pierre de Fermat has been called the greatest French mathematician of the seventeenth century (Eves,p.354). We have seen his contributions to calculus, the law of refraction, and most importantly to number theory. Fermat died on January 1 2, 1665 in Castres, France. Unfortunately Fermat's influence was not very great, because he was reluctant to publish his work. However, he is still remembered as a very great mathematician.