cp's OEIS Frontend

A076440 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).

%I A076440 #13 Feb 16 2025 08:32:47
%S A076440 1,2,10,30,38,46,122,126,138,142,146,150,154,166,170,190,194,214,222,
%T A076440 234,270,282,298,318,338,342,354,370,382,386,406,486,490,498,502,518,
%U A076440 546,550,566,574,582,586,594,638,666,678,686,694,710,726,730,734,746
%N A076440 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 A076440 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 A076440 R. K. Guy, Unsolved Problems in Number Theory, D9.
%D A076440 T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986.
%H A076440 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 A076440 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 A076440 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PillaisConjecture.html">Pillai's Conjecture</a>.
%Y A076440 Cf. A001597, A076438, A076439.
%K A076440 hard,nonn
%O A076440 1,2
%A A076440 _T. D. Noe_, Oct 12 2002