**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,^{ [1] } and in 1880 he published the first general solution^{ [2] } 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 classical mathematics, **analytic geometry**, also known as **coordinate geometry** or **Cartesian geometry**, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.

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 is represented by 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 traveled 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 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.

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.

**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 (posthumously) in 1676 and reprinted in 1729.

In mathematics, the **Fermat curve** is the algebraic curve in the complex projective plane defined in homogeneous coordinates (*X*:*Y*:*Z*) by the **Fermat equation**

In additive number theory, Fermat's theorem on sums of two squares states that an odd prime *p* can be expressed as:

In number theory, a **congruent number** is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property.

The **Beal conjecture** is the following conjecture in number theory:

In number theory, a branch of mathematics, a **Mirimanoff's congruence** is one of a collection of expressions in modular arithmetic which, if they hold, entail the truth of Fermat's Last Theorem. Since the theorem has now been proven, these are now of mainly historical significance, though the Mirimanoff polynomials are interesting in their own right. The theorem is due to Dmitry Mirimanoff.

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.

In number theory, a **congruum** is the difference between successive square numbers in an arithmetic progression of three squares. That is, if , , and are three square numbers that are equally spaced apart from each other, then the spacing between them, , is called a congruum.

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, published in 1670 among the works of Pierre de Fermat, soon after his death. It is the only complete proof given by Fermat. It has several equivalent formulations, one of which was stated in 1225 by Fibonacci. In its geometric forms, it states:

- ↑ Pepin T (1876). "Impossibilité de l'équation
*x*^{7}+*y*^{7}+*z*^{7}= 0".*C. R. Acad. Sci. Paris*.**82**: 676–679, 743–747. - ↑ Pepin T (1880). "Solution d'un Problème de Frenicle Sur Deux Triangles Rectangles".
*Atti Accad. Pont. Nuovi Lincei*.**33**: 284–289.

- 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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