%I #118 Jul 09 2025 04:34:17

%S 9,81,512,5041,14884,3805914951397,4902227890625,235260548044817,

%T 443689062789184,902576261010649

%N 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.

%C Probably finite.

%C 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]

%C From _Tomohiro Yamada_, Nov 19 2017: (Start)

%C 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).

%C Finiteness of this sequence would follow from the abc-conjecture.

%C 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)

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

%H Jean-François Alcover, <a href="/A214618/a214618.txt">Mathematica program.</a> [Recomputes the 6 parameters x,y,z and p,q,r from existing data].

%H H. Darmon and A. Granville, <a href="http://dx.doi.org/10.1112/blms/27.6.513">On the Equations z^m = F(x, y) and Ax^p + By^q = Cz^r</a> , Bull. London Math. Soc. 27 (1995), 513-543, available from <a href="http://www.dms.umontreal.ca/~andrew/PDF/superelliptic.pdf">the second author's page</a>.

%H R. Mauldin, <a href="http://www.ams.org/notices/199711/beal.pdf">A generalization of Fermat's Last Theorem: The Beal conjecture and prize problem</a>, Notices Am. Math. Soc. 44 (1997), no. 11, pp. 1436-1437.

%H Carl Pomerance, <a href="http://www.math.dartmouth.edu/~carlp/PDF/pcm0049.pdf">Computational Number Theory</a>

%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_559.htm">Puzzle 559. Mauldin / Tijdeman-Zagier Conjecture</a>

%H M. Waldschmidt, <a href="http://www.sms.edu.pk/wc2013.php">Lecture on the abc conjecture and some of its consequences</a>, Abdus Salam School of Mathematical Sciences (ASSMS), Lahore, 6th World Conference on 21st Century Mathematics 2013.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BealsConjecture.html">Beal's Conjecture</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fermat-CatalanConjecture.html">Fermat-Catalan Conjecture</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Beal%27s_conjecture">Beal's conjecture</a>

%e 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.

%e 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

%Y Cf. A001597.

%K hard,more,nonn

%O 1,1

%A _Arkadiusz Wesolowski_, Mar 06 2013