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Search: id:A076427
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| A076427 |
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Number of solutions to Pillai's equation a^x - b^y = n, with a>0, b>0, x>1, y>1 and n<=100. |
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3
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| 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
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OFFSET
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1,3
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COMMENT
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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.
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REFERENCES
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M. A. Bennett, "On some exponential equations of S.S. Pillai", Canad. J. Math. 53 (2001), 897-922.
R. K. Guy, Unsolved Problems in Number Theory, D9.
T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986.
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LINKS
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T. D. Noe, Solutions to Pillai's Equation for n<=100
Eric Weisstein's World of Mathematics, Pillai's Conjecture
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EXAMPLE
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a(4)=3 because there are 3 solutions: 4 = 2^3 - 2^2 = 6^2 - 2^5 = 5^3 - 11^2.
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CROSSREFS
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Cf. A001597, A074981.
Sequence in context: A056888 A111182 A079677 this_sequence A011024 A105855 A152954
Adjacent sequences: A076424 A076425 A076426 this_sequence A076428 A076429 A076430
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KEYWORD
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hard,nonn
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Oct 11 2002
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