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A214618
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Perfect powers z^r that can be written in the form x^p + y^q, where x, y, z are positive coprime integers and p, q, r are positive integers satisfying 1/p + 1/q + 1/r < 1.
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1
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OFFSET
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1,1
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COMMENTS
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Probably finite.
The Fermat-Catalan conjecture states that there are only finitely many terms. For the values of the six parameters x, y, z, p, q and r corresponding to the ten known terms, see the Weisstein article on this conjecture under Links. For each of the ten known terms, at least one of the exponents p, q and r is 2. A closely-related conjecture, the Tijdeman-Zagier conjecture (known more popularly as Beal's conjecture) is that there exists no set of three positive coprime integers x, y, z such that x^p + y^q = z^r where p, q, r are all integers greater than 2. The Beal problem, for which there is a $1,000,000 prize, is to find such a solution or to show that no such solution exists. See Mauldin (1997). - N. J. A. Sloane, Dec 22 2013 [Edited by Jon E. Schoenfield, Oct 03 2015]
In the Fermat-Calatan conjecture and Beal's conjecture, it must be required that the x, y, z are coprime. Otherwise, these conjectures would fail. For example, 2^n + 2^n = 2^(n+1). Moreover, for any integers a, b and n, z = a^n + b^n, x = az and y = bz, the equality x^n + y^n = z^(n+1) holds. There exist other "counterexamples" such as (3^3)^n + (2 * 3^n)^3 = 3^(3n+2) (derived from 1 + 2^3 = 3^2).
Finiteness of this sequence would follow from the abc-conjecture.
For each fixed A, B, C, p, q and r with 1/p + 1/q + 1/r < 1, the equation Ax^p + By^q = Cz^r has only finitely many coprime integer solutions x, y and z (H. Darmon, A. Granville). (End)
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REFERENCES
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Richard Crandall and Carl Pomerance, Prime Numbers - A Computational Perspective, Second Edition, Springer, 2005, ISBN 0-387-25282-7, pp. 416-417.
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LINKS
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Jean-François Alcover, Mathematica program. [Recomputes the 6 parameters x,y,z and p,q,r from existing data].
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EXAMPLE
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13^2 + 7^3 = 2^9 = 512. The numbers 13, 7, and 2 form a coprime set and 1/2 + 1/3 + 1/9 < 1. Therefore 512 is a term.
The factorizations of the known terms are 3^2, 3^4, 2^9, 71^2, 122^2, 15613^3, 65^7, 113^7, 21063928^2, 30042907^2. - N. J. A. Sloane, Dec 22 2013
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CROSSREFS
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KEYWORD
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hard,more,nonn,changed
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AUTHOR
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STATUS
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approved
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