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%I #8 Dec 11 2012 21:33:59
%S 1,2,10,30,38,46,122,126,138,142,146,150,154,166,170,190,194,214,222,
%T 234,270,282,298,318,338,342,354,370,382,386,406,486,490,498,502,518,
%U 546,550,566,574,582,586,594,638,666,678,686,694,710,726,730,734,746
%N 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).
%C 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.
%D R. K. Guy, Unsolved Problems in Number Theory, D9.
%D T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986.
%H M. E. Bennett, <a href="http://www.math.ubc.ca/~bennett/paper19.pdf">On Some Exponential Equations Of S. S. Pillai</a>,Canad. J. Math. 53 (2001), 897-922.
%H T. D. Noe, <a href="http://www.sspectra.com/Pillai1b.txt">Unique solutions to Pillai's Equation requiring an odd power for n<=1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PillaisConjecture.html">Pillai's Conjecture</a>
%Y Cf. A001597, A076438, A076439.
%K hard,nonn
%O 1,2
%A _T. D. Noe_, Oct 12 2002
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