Fermat's Theorem


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Fermat's theorem

[′fer‚mäz ‚thir·əm]
(mathematics)
The proposition that, if p is a prime number and a is a positive integer which is not divisible by p, then a p-1-1 is divisible by p.

Fermat’s Theorem

 

(or Fermat’s lesser theorem), a fundamental theorem of number theory. The theorem states that if p is a prime number and a is a whole number not divisible by p, then ap–1 – 1 is divisible by p—that is, ap–1 ≡ 1 (mod p). The theorem was set forth by P. Fermat without proof; the first proof was given by L. Euler.

References in periodicals archive ?
These two lectures are followed by a series of essays on artificial intelligence, Fermat's theorem, Euclid, and Hilbert.
For example, in a discussion about the solving of Fermat's theorem we read, "Two young mathematicians.
For example, he describes how Fermat's last theorem, first posited in the year 1630, remained unsolved until Andrew Wiles published his solution in 1995 while also explaining how work on Fermat's theorem led to the development of algebraic number theory and complex analysis.
Fermat's theorem had bedeviled mathematicians for 350 years.
While this sounds good on paper, twitchy, high- strung eggheads rarely project a good leader image, spending as much time as they do pondering Fermat's Theorem, muttering to themselves and collecting their toenail clippings in large jars for future cloning experiments they plan to get to just as soon as they finish calculating Pi to the 20 billionth decimal and cleaning up the garage.
Although this volume is considerably more linear than the two previous ones, genres are constantly mixed as autobiographical material is interlaced with erudition (on the history of mathematics), references to contemporary events, theoretical discussions (of Fermat's theorem, for example), essays (on the Field Prize), and vignettes of people Roubaud has known, some of them of importance in mathematics and intellectual history.
Since it will not, like Fermat's theorem, ever submit to proof, the Gogol problem persists as a peculiar challenge to the capacities of literary scholars.
Quoted in the June Scientific American, veteran mathematician Andre Weil of the Institute for Advanced Study said, "[T]o some extent, proving Fermat's theorem is like climbing Everest.
Just as 300 years of failed efforts to prove the hypothesis of Fermat's theorem did not demonstrate the falsity of that mathematical hypothesis, as has now been seen through its proof in the 1980s, a proof built upon the work of earlier "failures," so the falsity of the "communist hypothesis" has not been demonstrated by the perceived failures of successive self-declared socialist and communist regimes and movements, argues French philosopher Badiou.
And gradually, the potential for solving Fermat's theorem became linked with other questions in mathematics.
Some recent mathematical discoveries, however, have tied Fermat's theorem more closely to modern mathematics, suggesting a possible avenue to a proof.