The atlas's own conjecture-become-theorem precedent, carried here as a theorem rather than a conjecture because it is now proved, with both ends of its 358-year history on record: Pierre de Fermat's 1637 marginal claim to a proof too large for the margin to hold, and Andrew Wiles's corrected 1995 proof, produced with Richard Taylor after a gap found in Wiles's original 1993 announcement. No trace of any proof by Fermat himself survives, and mathematicians generally believe the elementary methods available in 1637 could not have proved the general case; Sophie Germain, among others, proved significant partial results toward it in the intervening centuries. Wiles's proof works by establishing enough of a separate, deep conjecture, the modularity theorem linking elliptic curves and modular forms, to force Fermat's claim as a consequence.
Facts
StatementNo three positive integers a, b and c can satisfy the equation a to the n plus b to the n equals c to the n, for any integer value of n strictly greater than two. 1 Proof YearWiles announced a proof in June 1993; reviewers found a gap the same year, and the corrected proof, with Richard Taylor, was published in 1995. Classification
Statement Form Statement Form Connections
Associated With
Fermat's Last Theorem is the special case of the Beal conjecture where the three exponents are forced equal, per Wikipedia's Relation to other conjectures section.
Source Beal Conjecture (Wikipedia)
Has Statement Form
In Branch
Additional Source Wikipedia: Fermat's Last TheoremWiles's general proof section
Source Encyclopaedia Britannica, Mathematics
Additional Source Wikipedia: Fermat's Last TheoremGeneralized Fermat equation
Named After
Posed By
Source Encyclopaedia Britannica, Mathematics
Additional Source Wikipedia: Fermat's Last TheoremHistory section
Proved By
Source Encyclopaedia Britannica, Mathematics
Additional Source Wikipedia: Fermat's Last TheoremLead section
Sources
1. Encyclopaedia Britannica, Mathematics
Fermat (Wiktionary)
Wikimedia FoundationPronunciation section, General AmericanQuote, Pronunciation section, General American
/ˈfɜɹmæt/, /ˈfɜɹmɑ/, /ˈfɛɹmɑ/
Beal Conjecture (Wikipedia)
Wikimedia FoundationAssociated With: Beal Conjecture, Relation to other conjectures sectionQuote, Associated With: Beal Conjecture, Relation to other conjectures section
Fermat's Last Theorem can be seen as a special case of the Beal conjecture restricted to x = y = z.
Wikipedia: Fermat's Last Theorem
Wikimedia FoundationPosed By: Pierre de Fermat, History section
Around 1637, Fermat wrote his Last Theorem in Latin in the margin of his copy of the Arithmetica next to Diophantus's sum-of-squares problem
Proved By: Andrew Wiles, Lead section
After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles and formally published in 1995.
In Branch: Number Theory, Generalized Fermat equation
The Beal conjecture states that there are no solutions to the generalized Fermat equation in positive integers a, b, c, m, n, k with a, b, and c being pairwise coprime and all of m, n, k being greater than 2. The Fermat-Catalan conjecture generalizes Fermat's last theorem with the ideas of the Catalan conjecture.
In Branch: Algebraic Number Theory, Wiles's general proof section
The proof's method of identification of a deformation ring with a Hecke algebra (now referred to as an R=T theorem) to prove modularity lifting theorems has been an influential development in algebraic number theory.
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