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Any solutions to the Beal conjecture will necessarily involve three terms all of which are 3-powerful numbers, i.e. numbers where the exponent of every prime factor is at least three. It is known that there are an infinite number of such sums involving coprime 3-powerful numbers; [ 11 ] however, such sums are rare.
The Beal conjecture, also known as the Mauldin conjecture [162] and the Tijdeman-Zagier conjecture, [163] [164] [165] 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. [166]
Beal's conjecture: for all integral solutions to + = where ,, >, all three numbers ,, must share some prime factor. Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture , per Tunnell's theorem ): determine precisely what rational numbers are congruent numbers .
The section Related examples contains this sentence: "Any solutions to the Beal conjecture will necessarily involve three terms all of which are ...." It is unfortunate to say a "solution" is to "the Beal conjecture". Each "solution" referred to here is a point (A,B,C) of the locus {(A,B,C) ∈ ℕ 3 | A x + B y = C z}.
The abc conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985. [ 1 ] [ 2 ] It is stated in terms of three positive integers a , b {\displaystyle a,b} and c {\displaystyle c} (hence the name) that are relatively prime and satisfy a ...
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 :
Fermat's Last Theorem, that a n + b n = c n has no solutions in positive integers with n > 2, is a special case of Beal's conjecture, that a x + b y = c z has no primitive solutions in positive integers with x, y, and z all greater than 2, specifically, the case of x = y = z.
For decades, the conjecture remained an important but unsolved problem in mathematics. Around 50 years after first being proposed, the conjecture was finally proven and renamed the modularity theorem, largely as a result of Andrew Wiles's work described below.