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Search: id:A076438
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| A076438 |
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n which appear to have a unique representation as the difference of two perfect powers; that is, there is only one solution to Pillai's equation a^x - b^y = n, with a>0, b>0, x>1, y>1. |
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+0
5
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| 1, 2, 10, 29, 30, 38, 43, 46, 52, 59, 122, 126, 138, 142, 146, 150, 154, 166, 170, 173, 181, 190, 194, 214, 222, 234, 263, 270, 282, 283, 298, 317, 318, 332, 338, 342, 347, 349, 354, 361, 370, 379, 382, 383, 386, 406, 419, 428, 436, 461, 467, 479, 484, 486
(list; graph; listen)
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OFFSET
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1,2
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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. See A076427 for the number of solutions for each n. Interestingly, the unique solutions (n,a,x,b,y) fall into two groups: (A076439) those in which x and y are even numbers, so that n is the difference of squares and (A076440) those requiring an odd power. This sequence was found by examining all perfect powers (A001597) less than 2^63-1. By examining a larger set of perfect powers, we may discover that some of these numbers do not have a unique representation.
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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, Unique solutions to Pillai's Equation for n<=1000
Eric Weisstein's World of Mathematics, Pillai's Conjecture
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CROSSREFS
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Cf. A001597, A076427, A076439, A076440.
Sequence in context: A124023 A127921 A106184 this_sequence A101561 A047112 A032250
Adjacent sequences: A076435 A076436 A076437 this_sequence A076439 A076440 A076441
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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 12 2002
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