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**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.

**Cluses** is a commune in the Haute-Savoie department in the Auvergne-Rhône-Alpes region in southeastern France.

**Haute-Savoie** is a department in the Auvergne-Rhône-Alpes region of Southeastern France, bordering both Switzerland and Italy. Its prefecture is Annecy. To the north is Lake Geneva and Switzerland; to the south and southeast are the Mont Blanc and Aravis mountain ranges. It holds it name from the Savoy historical region, as does the department of Savoie, located south of Haute-Savoie.

**Rome** is the capital city and a special *comune* of Italy. Rome also serves as the capital of the Lazio region. With 2,872,800 residents in 1,285 km^{2} (496.1 sq mi), it is also the country's most populated *comune*. It is the fourth most populous city in the European Union by population within city limits. It is the centre of the Metropolitan City of Rome, which has a population of 4,355,725 residents, thus making it the most populous metropolitan city in Italy. Rome is located in the central-western portion of the Italian Peninsula, within Lazio (Latium), along the shores of the Tiber. The Vatican City is an independent country inside the city boundaries of Rome, the only existing example of a country within a city: for this reason Rome has been often defined as capital of two states.

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

**Number theory** is a branch of pure mathematics devoted primarily to the study of the integers. 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 objects made out of integers or defined as generalizations of the integers.

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 to have infinitely many solutions since antiquity.

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

In mathematics, a **Diophantine equation** is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied. A **linear Diophantine equation** equates the sum of two or more monomials, each of degree 1 in one of the variables, to a constant. An **exponential Diophantine equation** is one in which exponents on terms can be unknowns.

**Pell's equation** is any Diophantine equation of the form

**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

**Pippin**, **Peppin**, **Pepin** or **Pipin** may refer to:

**Calculus of variations** is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.

**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.

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 **Frobenius pseudoprime** is a pseudoprime that passes a specific probable prime test described by Jon Grantham in a 1998 preprint and published in 2000. It has been studied by other authors for 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 is 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 number theory, a **congruum** is the difference between successive square numbers in an arithmetic progression of three squares. That is, if *x*^{2}, *y*^{2}, and *z*^{2} are three square numbers that are equally spaced apart from each other, then the spacing between them, *z*^{2} − *y*^{2} = *y*^{2} − *x*^{2}, 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**.

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 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 left 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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