# Pierre de Fermat

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## Fermat, Pierre de

(pyĕr də fĕrmä`), 1601–65, French mathematician. A magistrate whose avocation was mathematics, Fermat is known as a founder of modern number theory**number theory,**

branch of mathematics concerned with the properties of the integers (the numbers 0, 1, −1, 2, −2, 3, −3, …). An important area in number theory is the analysis of prime numbers.

**.....**Click the link for more information. and probability

**probability,**

in mathematics, assignment of a number as a measure of the "chance" that a given event will occur. There are certain important restrictions on such a probability measure.

**.....**Click the link for more information. theory. He also did much to establish coordinate geometry (see Cartesian coordinates

**Cartesian coordinates**

[for René Descartes], system for representing the relative positions of points in a plane or in space. In a plane, the point

*P*is specified by the pair of numbers (

*x,y*

**.....**Click the link for more information. ) and invented a number of methods for determining maxima and minima that were later of use to Newton in applying the calculus. He noted without proof, although he claimed to have discovered one, the assertion now known as

**Fermat's Last Theorem,**which states that the equation

*x*

^{n}+

*y*

^{n}=

*z*

^{n}, where

*x,*

*y,*

*z,*and

*n*are nonzero integers, has no solutions for

*n*that are greater than 2. Prizes were offered for a proof of this theorem, and attempted proofs resulted in many developments in the theory of numbers. British mathematician Andrew Wiles described a proof of the conjecture in 1993, but a gap in the proof required additional work, which was completed in 1994. However, Wiles's proof involved mathematical concepts that were unknown in Fermat's lifetime, so whether Fermat had a valid proof remains conjecture. In optics Fermat recognized that of all possible paths, light takes the path that takes the least time; this fundamental rule is known as Fermat's principle.

### Bibliography

See M. S. Mahoney, *The Mathematical Career of Pierre de Fermat 1601–1665* (2d rev. ed. 1994); A. D. Aczel, *Fermat's Last Theorem* (1996); S. Singh and J. Lynch, *Fermat's Enigma* (1998).

## Fermat, Pierre de

Born Aug. 17, 1601, in Beaumont-de-Lomagne; died Jan. 12, 1665, in Castres. French mathematician.

A lawyer by profession, Fermat became a councillor of the parliament of Toulouse in 1631. He produced a number of remarkable works; most of them were published posthumously in 1679 in his *Varia opera mathematica* (Various Mathematical Works), which was edited by his son. The results that Fermat obtained became known to scholars during his lifetime through correspondence and personal contact.

Fermat is one of the founders of the theory of numbers. Two famous theorems in this field bear his name: Fermat’s last theorem, also known as Fermat’s great theorem, and Fermat’s lesser theorem, which is usually called simply Fermat’s theorem. In geometry he developed the method of coordinates in a more systematic form than did R. Descartes; Fermat found the equations of the straight line and of second-degree curves and outlined a proof for the assertion that all second-degree curves are conies.

In the area of the method of infinitesimals, Fermat systematically studied the process of differentiation; he gave a general rule for the differentiation of powers and applied the rule to the differentiation of fractional powers. A method devised by him for finding extrema was of great importance in the development of modern methods of differential calculus. Before Fermat the rule for the integration of a power had been known for some special cases. He provided a general proof of the rule’s correctness and extended the rule to the cases of fractional and negative powers. Both of the fundamental processes of the method of infinitesimals thus received systematic development by Fermat. Like his contemporaries, however, he failed to notice the relation between the operations of differentiation and integration. This relation was established somewhat later in systematic form by G. von Leibniz and I. Newton.

Through his works Fermat had a great influence on the subsequent development of mathematics. In physics he set forth the fundamental law of geometrical optics now known as Fermat’s principle.

### WORKS

*Oeuvres*, vols. 1–4. Paris, 1891–1912.

### REFERENCES

Bourbaki, N.*Ocherki po istorii matematiki*[book 8]:

*Elementy matematiki*. Moscow, 1963. (Contains bibliography.) (Translated from French.)

*Istoriia matematiki s drevneishikh vremen do nachala XIX stoletiia*, vol. 2. Moscow, 1970.