A214618 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.

9, 81, 512, 5041, 14884, 3805914951397, 4902227890625, 235260548044817, 443689062789184, 902576261010649

Offset

1,1

Comments

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]

From Tomohiro Yamada, Nov 19 2017: (Start)

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)

References

Richard Crandall and Carl Pomerance, Prime Numbers - A Computational Perspective, Second Edition, Springer, 2005, ISBN 0-387-25282-7, pp. 416-417.

Links

Table of n, a(n) for n=1..10.

Jean-François Alcover, Mathematica program. [Recomputes the 6 parameters x,y,z and p,q,r from existing data].

H. Darmon and A. Granville, On the Equations z^m = F(x, y) and Ax^p + By^q = Cz^r , Bull. London Math. Soc. 27 (1995), 513-543, available from the second author's page.

R. Mauldin, A generalization of Fermat's Last Theorem: The Beal conjecture and prize problem, Notices Am. Math. Soc. 44 (1997), no. 11, pp. 1436-1437.

Carl Pomerance, Computational Number Theory

Carlos Rivera, Puzzle 559. Mauldin / Tijdeman-Zagier Conjecture

M. Waldschmidt, Lecture on the abc conjecture and some of its consequences, Abdus Salam School of Mathematical Sciences (ASSMS), Lahore, 6th World Conference on 21st Century Mathematics 2013.

Eric Weisstein's World of Mathematics, Beal's Conjecture

Eric Weisstein's World of Mathematics, Fermat-Catalan Conjecture

Wikipedia, Beal's conjecture

Example

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

Crossrefs

Cf. A001597.

Sequence in context: A208079 A223719 A270295 * A343462 A207009 A196986

Adjacent sequences: A214615 A214616 A214617 * A214619 A214620 A214621

Keyword

hard,more,nonn

Author

Arkadiusz Wesolowski, Mar 06 2013

Status

approved