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In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation a n + b n = c n for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions. [1]
The first of these (1 m + 2 3 = 3 2) is the only solution where one of a, b or c is 1, according to the Catalan conjecture, proven in 2002 by Preda Mihăilescu. While this case leads to infinitely many solutions of (1) (since one can pick any m for m > 6), these solutions only give a single triplet of values (a m, b n, c k).
Conjecture Field Comments Eponym(s) Cites 1/3–2/3 conjecture: order theory: n/a: 70 abc conjecture: number theory: ⇔Granville–Langevin conjecture, Vojta's conjecture in dimension 1 ⇒Erdős–Woods conjecture, Fermat–Catalan conjecture Formulated by David Masser and Joseph Oesterlé. [1] Proof claimed in 2012 by Shinichi Mochizuki: n/a ...
Fermat's last theorem Fermat's last theorem, one of the most famous and difficult to prove theorems in number theory, states that for any integer n > 2, the equation a n + b n = c n has no positive integer solutions. Fermat's little theorem Fermat's little theorem field extension A field extension L/K is a pair of fields K and L such that K is ...
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. [1] [2] [3] Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to ...
Fermat's Last Theorem states that no three positive integers (a, b, c) can satisfy the equation a n + b n = c n for any integer value of n greater than 2. (For n equal to 1, the equation is a linear equation and has a solution for every possible a and b.
The Fermat–Catalan conjecture is an open conjecture dealing with such cases (the condition of this conjecture is that the sum of the reciprocals is less than 1). If we allow at most one of the exponents to be 2, then there may be only finitely many solutions (except the case 1 m + 2 3 = 3 2 {\displaystyle 1^{m}+2^{3}=3^{2}} ).
In number theory, Euler's conjecture is a disproved conjecture related to Fermat's Last Theorem.It was proposed by Leonhard Euler in 1769. It states that for all integers n and k greater than 1, if the sum of n many k th powers of positive integers is itself a k th power, then n is greater than or equal to k: