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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.
Germain tried unsuccessfully to prove the first case of Fermat's Last Theorem for all even exponents, specifically for n = 2p, which was proved by Guy Terjanian in 1977. [129] In 1985, Leonard Adleman, Roger Heath-Brown and Étienne Fouvry proved that the first case of Fermat's Last Theorem holds for infinitely many odd primes p. [130]
Fermat's Last Theorem, formulated in 1637, states that no three positive integers a, b, and c can satisfy the equation + = if n is an integer greater than two (n > 2).. Over time, this simple assertion became one of the most famous unproved claims in mathematics.
Kummer improved this further in 1857 by showing that for the "first case" of Fermat's Last Theorem (see Sophie Germain's theorem) it is sufficient to establish that either (p, p − 3) or (p, p − 5) fails to be an irregular pair. ((p, 2k) is an irregular pair when p is irregular due to a certain condition described below being realized at 2k.)
A typical such equation is the equation of Fermat's Last Theorem x d + y d − z d = 0. {\displaystyle x^{d}+y^{d}-z^{d}=0.} As a homogeneous polynomial in n indeterminates defines a hypersurface in the projective space of dimension n − 1 , solving a homogeneous Diophantine equation is the same as finding the rational points of a projective ...
The works of the 17th-century mathematician Pierre de Fermat engendered many theorems. Fermat's theorem may refer to one of the following theorems: Fermat's Last Theorem, about integer solutions to a n + b n = c n; Fermat's little theorem, a property of prime numbers; Fermat's theorem on sums of two squares, about primes expressible as a sum of ...
The equation + = has no solutions in positive integers and pairwise coprime integers A, B, C if x, y, z ≥ 3. The conjecture was formulated in 1993 by Andrew Beal, a banker and amateur mathematician, while investigating generalizations of Fermat's Last Theorem.
In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction [1] used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. [2]