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A076439
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Numbers n which appear to have a unique representation as the difference of two perfect powers and those powers are both 2; 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 x=y=2.
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2
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29, 43, 52, 59, 173, 181, 263, 283, 317, 332, 347, 349, 361, 379, 383, 419, 428, 436, 461, 467, 479, 484, 491, 509, 523, 529, 569, 571, 607, 613, 619, 641, 643, 653, 661, 677, 691, 709, 733, 773, 787, 788, 811, 827, 839, 853, 877, 881, 883, 907, 911, 941
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| There are two types of unique solutions. See A076438 for the general case. The n for which the unique solution can be written as n = a^2 - b^2 seems to have the following properties: (1) b = a-1 for odd n and b = a-2 for even n and (2) n = 4^r p^s, where r is in {0,1}, p is an odd prime and s is in {1,2). 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
| 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
| Eric Weisstein's World of Mathematics, Pillai's Conjecture
T. D. Noe, Unique solutions to Pillai's Equation requiring only squares for n<=1000
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CROSSREFS
| Cf. A001597, A076438, A076440.
Sequence in context: A086149 A066502 A125870 * A168474 A162357 A173967
Adjacent sequences: A076436 A076437 A076438 * A076440 A076441 A076442
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KEYWORD
| hard,nonn
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AUTHOR
| T. D. Noe (noe(AT)sspectra.com), Oct 12 2002
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