Construct · 1 fields · 3 citations

Fermat's Last Theorem: a standalone construct

One name. One construct. A landmark on the map.

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Fermat'sLast Theorem1 CONSTRUCTFermat's Last ThmFERMAT · a^n+b^n=c^n

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Number theory

The number-theory result that a^n + b^n = c^n has no positive-integer solutions for n greater than 2

Pierre de Fermat first stated the claim around 1637 in the margin of his copy of Diophantus's Arithmetica, asserting that no three positive whole numbers a, b, and c can satisfy the equation a^n + b^n = c^n for any whole-number exponent n greater than 2, and adding that he had found a truly marvelous proof that the margin was too narrow to contain; the note was published by his son Samuel de Fermat in 1670. For more than three centuries the statement resisted proof and became one of the most famous open problems in mathematics, checked for many individual exponents but never settled in general. Andrew Wiles gave the first complete proof in 1995, published as Modular elliptic curves and Fermat's Last Theorem in the Annals of Mathematics, by establishing enough of the modularity theorem, the deep and unexpected link between elliptic curves and modular forms (two very different kinds of mathematical object), to rule out any counterexample; a companion paper with Richard Taylor closed a gap in the argument. What makes it a landmark is that a plain-looking claim about ordinary whole numbers turned out to require, and to help drive, some of the deepest machinery in modern number theory to settle.[1][2][3]

Unit
A theorem of number theory: a universal statement that a stated equation has no solutions in positive integers
Recovery means
Not applicable
Measured by
Established by mathematical proof; the general result was proved in 1995 via the modularity theorem, after centuries of verified special cases for individual exponents

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Nearby constructs

proof2 fields · 5 citationsjingle · one word, two constructsargument3 fields · 3 citationsjingle · one word, three constructsclosed3 fields · 3 citationsjingle · one word, three constructs

References

  1. Fermat, P. de. (1670). Observationes D. Petri de Fermat. In Diophanti Alexandrini Arithmeticorum libri sex (S. de Fermat, Ed.). Bernard Bosc. (Marginal note originally written c. 1637)
  2. Wiles, A. (1995). Modular elliptic curves and Fermat's Last Theorem. Annals of Mathematics, 141(3), 443–551. https://doi.org/10.2307/2118559
  3. Taylor, R., & Wiles, A. (1995). Ring-theoretic properties of certain Hecke algebras. Annals of Mathematics, 141(3), 553–572. https://doi.org/10.2307/2118560

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