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%I #25 Aug 16 2021 14:24:07
%S 1,1,2,3,2,0,5,3,4,1,4,2,3,0,3,3,7,3,5,2,2,2,4,5,2,3,3,7,1,1,2,4,2,0,
%T 3,2,3,1,4,4,3,0,1,3,4,1,6,4,3,0,2,1,2,2,3,4,3,0,1,4,2,0,4,4,4,0,2,5,
%U 2,0,4,4,6,2,3,3,2,0,4,4,4,0,2,2,2,0,3,3,6,0,3,4,4,2,4,5,3,2,4,10
%N Number of solutions to Pillai's equation a^x - b^y = n, with a>0, b>0, x>1, y>1.
%C This is the classic Diophantine equation of S. S. Pillai, who conjectured that there are only a finite number of solutions for each n. A generalization of Catalan's conjecture that a^x-b^y=1 has only one solution. For n <=100, a total of 274 solutions were found for perfect powers less than 10^12. No additional solutions were found for perfect powers < 10^18.
%D R. K. Guy, Unsolved Problems in Number Theory, D9.
%D T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986.
%H T. D. Noe, <a href="http://www.sspectra.com/Pillai.txt">Solutions to Pillai's Equation for n<=100</a>
%H Dorin Andrica and Ovidiu Bagdasar, <a href="https://doi.org/10.1007/s11139-021-00418-7">On k-partitions of multisets with equal sums</a>, The Ramanujan J. (2021) Vol. 55, 421-435.
%H Dorin Andrica and George C. Ţurkaş, <a href="https://doi.org/10.24193/subbmath.2019.3.06">An elliptic Diophantine equation from the study of partitions</a>, Stud. Univ. Babeş-Bolyai Math. (2019) Vol. 64, No. 3, 349-356.
%H M. A. Bennett, <a href="http://www.math.ubc.ca/~bennett/B-CJM-Pillai.pdf">On some exponential equations of S. S. Pillai</a>, Canad. J. Math. 53 (2001), 897-922.
%H Dana Mackenzie, <a href="http://math.colgate.edu/~integers/s33/s33.Abstract.html">2184: An absurd (and adsurd) tale</a>, Integers (2018) 18, Article #A33.
%H Roswitha Rissner and Daniel Windisch, <a href="https://arxiv.org/abs/2009.02322">Absolute irreducibility of the binomial polynomials</a>, arXiv:2009.02322 [math.AC], 2020.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PillaisConjecture.html">Pillai's Conjecture</a>
%e a(4)=3 because there are 3 solutions: 4 = 2^3 - 2^2 = 6^2 - 2^5 = 5^3 - 11^2.
%Y Cf. A189117, A001597, A074981.
%K hard,nonn
%O 1,3
%A _T. D. Noe_, Oct 11 2002
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