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A076427 Number of solutions to Pillai's equation a^x - b^y = n, with a>0, b>0, x>1, y>1. 5


%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 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 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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