A happy identity

The statement that secured Bézout's place in history takes either the form of the Bachet–Bézout theorem (given two integers a and b, a and b are coprime if and only if there exist two integers x and y such that ax + by = 1), or that of an identity (there exist two integers x and y such that ax + by = d, where d is the greatest common divisor of a and b). The identity follows easily from the theorem. Yet the result was no "scoop" for either of the two mathematicians whose names are associated with it: the first proved it in his own way, and the second generalized it.

Bachet, the forerunner

Bachet de Méziriac was equally at home with poetry, ancient languages and mathematics, but today he is known chiefly for his Problèmes plaisants et délectables qui se font par les nombres (see box) and the solving methods he presented in it.

Bachet, a humanist mathematician

Claude-Gaspar Bachet de Méziriac, who studied with the Jesuits and even belonged to the order for a year, lived the life of an aristocrat on his estate in Bourg-en-Bresse and mixed with literary figures such as Vaugelas, with whom he composed poems in Italian. He was among the first members of the newly founded Académie française. From 1634, he occupied its thirteenth seat, later held by many other illustrious figures: Jean Racine (in 1672), Pierre Loti (in 1891), Paul Claudel (in 1946), Maurice Schumann (in 1974) and Pierre Messmer (in 1999). Today, Simone Weil has held the seat since 2010.

Bachet was not strictly speaking a mathematician, but stated and proved the first Bézout theorem on coprime integers. Furthermore, he remains known as the first author to write a book of mathematical recreations in French; it has frequently been reissued and widely drawn upon since it first appeared in 1612. A skilful mathematician, he also wrote a Latin manuscript entitled Elementorum Arithmeticorum Libri (Éléments d'arithmétique). A highly accomplished scholar, he translated Diophantus's Arithmetica into Latin and enriched it with numerous notes of his own. Indeed, a copy of this translation bears Fermat's annotation claiming that he had a complete proof of his famous last theorem...

The breadth of his learning made him the very model of the cultivated gentleman and the 17^th-century humanist.

The idea behind the book was innovative: mathematical recreations, usually relegated to an appendix, were here the book's very subject. The author explained his own, often original problem-solving methods in detail. Rather than amusing problems used to illustrate serious theorems, these were recreational problems that prompted a search for mathematics useful for solving them.

Bachet, the forerunner

Comments

Join the discussion

On the same theme

Bachet and Bézout: a winning mathematical duo

The Chinese remainder theorem

Billiards and water-pouring problems

From Bézout's theorem for polynomials to the intersection of conics

Comments

Join the discussion