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Numbers k which appear to have a unique representation as the difference of two perfect powers where one of those powers is odd; that is, there is only one solution to Pillai's equation a^x - b^y = k, 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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%I #12 Jul 15 2024 02:08:12

%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 Numbers k which appear to have a unique representation as the difference of two perfect powers where one of those powers is odd; that is, there is only one solution to Pillai's equation a^x - b^y = k, 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