**Jean François Théophile Pépin** (14 May 1826 – 3 April 1904) was a French mathematician.

Born in Cluses, Haute-Savoie, he became a Jesuit in 1846, and from 1850 to 1856 and from 1862 to 1871 he was Professor of Mathematics at various Jesuit colleges. He was appointed Professor of Canon Law in 1873, moving to Rome in 1880. He died in Lyon at the age of 77.

His work centred on number theory. In 1876 he found a new proof of Fermat's Last Theorem for *n* = 7, and in 1880 he published the first general solution to Frénicle de Bessy's problem

*x*^{2}+*y*^{2}=*z*^{2},*x*^{2}=*u*^{2}+*v*^{2},*x*−*y*=*u*−*v*.

He also gave his name to Pépin's test, a test of primality for Fermat numbers.

In mathematics, a **Diophantine equation** is a polynomial equation, usually involving two or more unknowns, such that the only solutions of interest are the integer ones. A **linear Diophantine equation** equates to a constant the sum of two or more monomials, each of degree one. An **exponential Diophantine equation** is one in which unknowns can appear in exponents.

**Number theory** is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers or defined as generalizations of the integers.

**Pell's equation**, also called the **Pell–Fermat equation**, is any Diophantine equation of the form where *n* is a given positive nonsquare integer and integer solutions are sought for *x* and *y*. In Cartesian coordinates, the equation has the form of a hyperbola; solutions occur wherever the curve passes through a point whose *x* and *y* coordinates are both integers, such as the trivial solution with *x* = 1 and *y* = 0. Joseph Louis Lagrange proved that, as long as *n* is not a perfect square, Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accurately approximate the square root of *n* by rational numbers of the form *x*/*y*.

**Fermat's principle**, also known as the **principle of least time**, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the path that can be traversed in the least time. In order to be true in all cases, this statement must be weakened by replacing the "least" time with a time that is "stationary" with respect to variations of the path — so that a deviation in the path causes, at most, a *second-order* change in the traversal time. To put it loosely, a ray path is surrounded by close paths that can be traversed in *very* close times. It can be shown that this technical definition corresponds to more intuitive notions of a ray, such as a line of sight or the path of a narrow beam.

**Fermat's little theorem** states that if p is a prime number, then for any integer a, the number *a*^{p} − *a* is an integer multiple of p. In the notation of modular arithmetic, this is expressed as

In mathematics, a **Fermat number**, named after Pierre de Fermat, who first studied them, is a positive integer of the form

In number theory, a **Wieferich prime** is a prime number *p* such that *p*^{2} divides 2^{p − 1} − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime *p* divides 2^{p − 1} − 1. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's last theorem, at which time both of Fermat's theorems were already well known to mathematicians.

**Bernard Frénicle de Bessy**, was a French mathematician born in Paris, who wrote numerous mathematical papers, mainly in number theory and combinatorics. He is best remembered for *Des quarrez ou tables magiques*, a treatise on magic squares published posthumously in 1693, in which he described all 880 essentially different normal magic squares of order 4. The Frénicle standard form, a standard representation of magic squares, is named after him. He solved many problems created by Fermat and also discovered the cube property of the number 1729, later referred to as a taxicab number. He is also remembered for his treatise *Traité des triangles rectangles en nombres* published in 1676.

**Arc length** is the distance between two points along a section of a curve.

**Fermat's factorization method**, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares:

In number theory, a **Frobenius pseudoprime** is a pseudoprime, whose definition was inspired by the quadratic Frobenius test described by Jon Grantham in a 1998 preprint and published in 2000. Frobenius pseudoprimes can be defined with respect to polynomials of degree at least 2, but they have been most extensively studied in the case of quadratic polynomials.

A **Pierpont prime** is a prime number of the form

In geometry, the **Fermat point** of a triangle, also called the **Torricelli point** or **Fermat–Torricelli point**, is a point such that the total distance from the three vertices of the triangle to the point is the minimum possible. It is so named because this problem was first raised by Fermat in a private letter to Evangelista Torricelli, who solved it.

In geometry, the **folium of Descartes** is an algebraic curve defined by the equation

In mathematics, **Pépin's test** is a primality test, which can be used to determine whether a Fermat number is prime. It is a variant of Proth's test. The test is named for a French mathematician, Théophile Pépin.

The **geometric median** of a discrete set of sample points in a Euclidean space is the point minimizing the sum of distances to the sample points. This generalizes the median, which has the property of minimizing the sum of distances for one-dimensional data, and provides a central tendency in higher dimensions. It is also known as the **1-median**, **spatial median**, **Euclidean minisum point**, or **Torricelli point**.

In number theory, **Fermat's Last Theorem** states that no three positive integers *a*, *b*, and *c* satisfy the equation *a*^{n} + *b*^{n} = *c*^{n} for any integer value of *n* greater than 2. The cases *n* = 1 and *n* = 2 have been known since antiquity to have infinitely many solutions.

Fermat's Last Theorem is a theorem in number theory, originally stated by Pierre de Fermat in 1637 and proved by Andrew Wiles in 1995. The statement of the theorem involves an integer exponent *n* larger than 2. In the centuries following the initial statement of the result and before its general proof, various proofs were devised for particular values of the exponent *n*. Several of these proofs are described below, including Fermat's proof in the case *n* = 4, which is an early example of the method of infinite descent.

**Fermat's right triangle theorem** is a non-existence proof in number theory, the only complete proof given by Pierre de Fermat. It has several equivalent formulations:

**Adequality** is a technique developed by Pierre de Fermat in his treatise *Methodus ad disquirendam maximam et minimam* to calculate maxima and minima of functions, tangents to curves, area, center of mass, least action, and other problems in calculus. According to André Weil, Fermat "introduces the technical term adaequalitas, adaequare, etc., which he says he has borrowed from Diophantus. As Diophantus V.11 shows, it means an approximate equality, and this is indeed how Fermat explains the word in one of his later writings.". Diophantus coined the word παρισότης (*parisotēs*) to refer to an approximate equality. Claude Gaspard Bachet de Méziriac translated Diophantus's Greek word into Latin as *adaequalitas*. Paul Tannery's French translation of Fermat’s Latin treatises on maxima and minima used the words *adéquation* and *adégaler*.

- Franz Lemmermeyer. "A Note on Pépin's counter examples to the Hasse principle for curves of genus 1."
*Abh. Math. Sem. Hamburg*69 (1999), 335–345.

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