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To illustrate, the solution + = has bases with a common factor of 3, the solution + = has bases with a common factor of 7, and + = + has bases with a common factor of 2. Indeed the equation has infinitely many solutions where the bases share a common factor, including generalizations of the above three examples, respectively
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.
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}.
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 :
⇐generalized Riemann hypothesis [2] ⇐Selberg conjecture B [3] Emil Artin: 325 Bateman–Horn conjecture: number theory: Paul T. Bateman and Roger Horn: 245 Baum–Connes conjecture: operator K-theory: ⇒Gromov-Lawson-Rosenberg conjecture [4] ⇒Kaplansky-Kadison conjecture [4] ⇒Novikov conjecture [4] Paul Baum and Alain Connes: 2670 Beal ...
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 ...
The conjecture is named after Paul Erdős and Ernst G. Straus, who formulated it in 1948, but it is connected to much more ancient mathematics; sums of unit fractions, like the one in this problem, are known as Egyptian fractions, because of their use in ancient Egyptian mathematics. The Erdős–Straus conjecture is one of many conjectures by ...
In 1966, a counterexample to Euler's sum of powers conjecture was found by Leon J. Lander and Thomas R. Parkin for k = 5: [1] 27 5 + 84 5 + 110 5 + 133 5 = 144 5. In subsequent years, further counterexamples were found, including for k = 4. The latter disproved the more specific Euler quartic conjecture, namely that a 4 + b 4 + c 4 = d 4 has no ...