David Wasserman ma163sar Written 6/8/95 Presented 6/9/95
Pierre de Fermat (1630 - 1665)
Pierre de Fermat was born in 1601 and died in 1665. By
profession he was a lawyer and jurist, and he studied mathematics in
his spare time. He is mostly remembered for his work in number
theory. He also worked in geometry. Fermat was a contemporary of
Descartes, and they both independently discovered analytic geometry.|
Fermat's most famous result, his so-called "Last Theorem," was found written in the margin of his copy of Diophantus' "Arithmetica." He wrote that he had a proof of the theorem, but the proof was not found in his effects. Also, in letters to other mathematicians, he frequently claimed to have proved results, but did not provide the proofs. He wanted to let others have the pleasure of discovering the proof, but this practice alienated some of his correspondents. Sometimes he provided proofs that were incomplete. This also annoyed his correspondents, who pressed him for more details.
Some of the gaps in these proofs were due to the fact that Fermat expected his readers to be familiar with certain works of François Viete (1540-1603) and the ancient Greek geometers, and to be able to apply the correct theorems from these works. Also, Fermat was strongly influenced by Viete, who revived interest in Greek analysis. The ancient Greeks divided their geometric arguments into two categories, analysis and synthesis. Synthesis was what we now call proof. Analysis meant assuming the proposition in question, and deducing from it something already known. This is not a logically valid way to verify the proposition. However, many theorems of geometry have converses that are also theorems, so analysis often produced correct results, and it worked often enough that it became common practice. Fermat recognized the need for synthesis, but he would often give an analysis of a theorem, and then state that it could easily be converted to a synthesis.
Fermat enjoyed producing his results, but was not willing to do the clean-up work required to make them suitable for publication. When others published his results, he would not allow them to put his name on them, because he wanted to avoid controversy. The following is a sample of theorems that Fermat did leave proofs of, written in modern notation and terminology.
Theorem: If a right triangle has integral sides, its area is not the square of an integer.
The proof is to long to include here. It can easily be modified to a proof that no fourth power is a sum of two fourth powers, a special case of the Last Theorem.
Theorem: If points A and B and a real number k are given, then the set of all points D, such that
Proof: Let C be the point on line AB such that
Theorem: Let a spiral be described by the polar equation
Proof: By solving for (R - r), we see that the section of the radius of the circle that lies outside the spiral, i.e., R - r, grows as the square of theta. Let the sector of angle alpha be split into N equal sectors. For each sector there is a circular arc inscribed in the spiral, and also a circumscribed circular arc. The segment between the circle and the spiral at the end of the ith sector is to that at the end of the (i - 1)th as
Theorem: No number of the form 4k - 1, where k is an integer, is a sum of two squares.
Proof: A square of an even number is divisible by 4. A square of an odd number 2h + 1 is 4h2 + 4h + 1 = 4k + 1. Therefore 4k - 1 is not a square. Suppose
Problem: Find the maximum value of bx2 - x3 in the first quadrant, where b is any positive number.
Solution: Let c be any value between 0 and the maximum value. The equation bx2 - x3 = c has two positive roots. Call them x and y. Then
This solution is correct but there is an flaw in the derivation.
Exercise: Find the flaw in the derivation.
Source: Mahoney, Michael Sean, "The Mathematical Career of Pierre de Fermat."