Pierre de Fermat

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Biography

Pierre de Fermat was born in Beaumont-de-Lomagne near Toulouse in 1607. He attended the University of Orléans from 1623 and received a bachelor in civil law in 1626, before moving to Bordeaux. In Bordeaux, he began his first serious mathematical researches, and he showed most of his work through letters to friends; in some of said letters, he explored many of the fundamental concepts of calculus before Newton or Leibniz.

Fermat was a trained lawyer making mathematics more of a hobby than a profession, which is why he never published his work. Regardless, he made important contributions to analytical geometry, probability, number theory and calculus. According to Peter L. Bernstein, in his 1996 book "Against the Gods", Fermat "was a mathematician of rare power. He was an independent inventor of analytic geometry, he contributed to the early development of calculus, he did research on the weight of the earth, and he worked on light refraction and optics. In the course of what turned out to be an extended correspondence with Blaise Pascal, he made a significant contribution to the theory of probability. But Fermat's crowning achievement was in the theory of numbers."

With his gift for number relations and his ability to find proofs for many of his theorems, Fermat essentially created the modern theory of numbers.

Pierre de Fermat died on January 12, 1665, at Castres, in the present-day department of Tarn. The oldest and most prestigious high school in Toulouse is named after him: the Lycée Pierre-de-Fermat. French sculptor Théophile Barrau made a marble statue named "Hommage à Pierre Fermat" as a tribute to Fermat, now at the Capitole de Toulouse.

Fermat's contributions to Discrete Mathematics

In contribution to number theory, Fermat studied Pell's equation, perfect numbers, amicable numbers and what would later become Fermat numbers.

It was while researching perfect numbers that he discovered Fermat's little theorem. He invented a factorization method—Fermat's factorization method—and popularized the proof by infinite descent, which he used to prove Fermat's right triangle theorem which includes as a corollary Fermat's Last Theorem for the case n = 4.

Fermat developed the two-square theorem, and the polygonal number theorem, which states that each number is a sum of three triangular numbers, four square numbers, five pentagonal numbers, and so on.

A. Fermat numbers

A Fermat number is a positive integer in the form:

where n is a non-negative integer.

The first few Fermat numbers are:

3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ... (sequence A000215 in the OEIS).

B. Fermat's little theorem

Fermat's little theorem states that if p is a prime number, then for any integer a, the number

is an integer multiple of p. In the notation of modular arithmetic, this is expressed as

For example, if a = 2 and p = 7, then 2^7 = 128, and 128 − 2 = 126 = 7 × 18 is an integer multiple of 7.

If a is not divisible by p, that is, if a is coprime to p, then Fermat's little theorem is equivalent to the statement that

is an integer multiple of p, or in symbols

For example, if a = 2 and p = 7, then 26 = 64, and 64 − 1 = 63 = 7 × 9 is thus a multiple of 7.

C. Fermat's last theorem

Additionally, Fermat's Last Theorem (sometimes called Fermat's conjecture) states that

"no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2."

Although other statements claimed by Fermat without proof were subsequently proven by others and credited as theorems of Fermat (for example, Fermat's theorem on sums of two squares), Fermat's Last Theorem resisted proof, leading to doubt that Fermat ever had a correct proof. Consequently, the proposition became known as a conjecture rather than a theorem.

After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles and formally published in 1995. It was described as a "stunning advance" in the citation for Wiles's Abel Prize award in 2016.

The unsolved problem stimulated the development of algebraic number theory in the 19th and 20th centuries. It is among the most notable theorems in the history of mathematics and prior to its proof was in the Guinness Book of World Recordsas the "most difficult mathematical problem", in part because the theorem has the largest number of unsuccessful proofs.

D. Method of infinite descent

In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. It is a method which relies on the well-ordering principle, and is often used to show that a given equation, such as a Diophantine equation, has no solutions.

Using a metaphor of climbing down a ladder, if a higher rung cannot be reached without first reaching a lower rung coupled with the notion that no lowest rung exists, then no rung can ever be reached.

Although Euclid first makes use of it in his "Elements", the method of infinite descent is attributed to Pierre de Fermat for stating it explicitly, in a letter to Christian Huygens : "I have finally organized this according to my method and shown that if a given number is not of this nature there will be a smaller number which also is not, then a third less than the second, etc., to infinity, from which one infers that all numbers are of this nature."

Reflections on the constructed nature of the digital world.

Finding sources on Pierre de Fermat was not difficult, and information on his contributions to mathematics was consistent. Seeing as he was one of the two leading mathematicians in the first half of the 16th century, records of his works are both detailed and plenty. A lot of the documentation of his work was sourced from his correspondence with other mathematicians. Still, despite his popularity at the time, his work was rarely published because Fermat did not really want to put his work in a polished form; most of his published work was put out by others.

The only inconsistency I could find does not have to do with his work, but with his personal life; most sources give Fermat's birth year as 1601 but recent research suggests that this was the year his half-brother called Piere (who died before Pierre was born) was born and, working backwards from the stated age at death, gives 1607 as his birth year.

References

1. “Fermat’s Theorem.” Encyclopædia Britannica, Encyclopædia Britannica, inc., Accessed Jan. 2024, Fermat's theorem.

2. “Pierre de Fermat.” Wikipedia, Wikimedia Foundation, Accessed Jan. 2024, Pierre de Fermat.

3. "Pierre de Fermat". “Complete Dictionary of Scientific Biography", Encyclopedia.com, Accessed Feb. 2024, Fermat, Pierre de.

4. “A000215.” OEIS, oeis.org/A000215. Accessed Feb. 2024, Fermat numbers.

5. “Pierre Fermat - Biography.” Maths History, mathshistory.st-andrews.ac.uk/Biographies/Fermat/. Accessed Feb. 2024, Pierre Fermat - Biography.

6. “Fermat’s Method of Infinite Descent.” Brilliant Math & Science Wiki, brilliant.org/wiki/general-diophantine-equations-fermats-method-of/. Accessed Feb. 2024, Fermat's Method of Infinite Descent.