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Search: id:A076440
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| A076440 |
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n which appear to have a unique representation as the difference of two perfect powers and one of those powers is odd; 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 and that solution has odd x or odd y (or both odd). |
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+0
4
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| 1, 2, 10, 30, 38, 46, 122, 126, 138, 142, 146, 150, 154, 166, 170, 190, 194, 214, 222, 234, 270, 282, 298, 318, 338, 342, 354, 370, 382, 386, 406, 486, 490, 498, 502, 518, 546, 550, 566, 574, 582, 586, 594, 638, 666, 678, 686, 694, 710, 726, 730, 734, 746
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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There are two types of unique solutions. See A076438 for the general case. 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 requiring an odd power for n<=1000
Eric Weisstein's World of Mathematics, Pillai's Conjecture
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CROSSREFS
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Cf. A001597, A076438, A076439.
Sequence in context: A098425 A098408 A063564 this_sequence A047198 A162524 A065137
Adjacent sequences: A076437 A076438 A076439 this_sequence A076441 A076442 A076443
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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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