Bézout's identity
Identidad de Bézout
- Date
- 1624 (Bachet) / 1779 (Bézout) · confidence medium
- Place
- Lyon, France
- Credited to
- Claude Gaspard Bachet de Méziriac (for integers); Étienne Bézout generalized the result to polynomials in 1779
- Field
- Knowledge & symbols
⚠️ Disputed: the sources disagree.
What was discovered
Bachet de Méziriac showed that for two coprime numbers there always exist integers that, multiplied by each and added, give 1, a key result for solving arithmetic problems with Euclid's algorithm. Étienne Bézout generalized the idea to polynomials more than a century and a half later, and ended up lending his name to the result even though it was already implicit in Euclid's Elements.
Historian Andrew Granville has noted that attributing the result to Bézout is a later (Bourbaki) tradition, since the integer case was implicit in Euclid and it was Bachet who first stated it explicitly in 1624.
What it led to
- Extended Euclidean algorithm
- Modern number theory
- RSA cryptography (modular arithmetic)