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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
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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. REFERENCES R. K. Guy, Unsolved Problems in Number Theory, D9. T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986. LINKS T. D. Noe, Solutions to Pillai's Equation for n<=100 M. A. Bennett, On some exponential equations of S. S. Pillai, Canad. J. Math. 53 (2001), 897-922. Dana Mackenzie, 2184: An absurd (and adsurd) tale, Integers (2018) 18, Article #A33. Eric Weisstein's World of Mathematics, Pillai's Conjecture EXAMPLE a(4)=3 because there are 3 solutions: 4 = 2^3 - 2^2 = 6^2 - 2^5 = 5^3 - 11^2. CROSSREFS Cf. A189117, A001597, A074981. Sequence in context: A286297 A111182 A178142 * A284152 A011024 A105855 Adjacent sequences:  A076424 A076425 A076426 * A076428 A076429 A076430 KEYWORD hard,nonn AUTHOR T. D. Noe, Oct 11 2002 STATUS approved

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Last modified June 16 19:43 EDT 2019. Contains 324155 sequences. (Running on oeis4.)