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%I #17 Jul 15 2024 02:08:05
%S 1,2,10,29,30,38,43,46,52,59,122,126,138,142,146,150,154,166,170,173,
%T 181,190,194,214,222,234,263,270,282,283,298,317,318,332,338,342,347,
%U 349,354,361,370,379,382,383,386,406,419,428,436,461,467,479,484,486
%N Numbers k which appear to have a unique representation as the difference of two perfect powers; 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.
%C This is the classic Diophantine equation of S. S. Pillai, who conjectured that there are only a finite number of solutions for each k. A generalization of Catalan's conjecture that a^x - b^y = 1 has only one solution. See A076427 for the number of solutions for each k. Interestingly, the unique solutions (k,a,x,b,y) fall into two groups: (A076439) those in which x and y are even numbers, so that k is the difference of squares, and (A076440) those requiring an odd power. 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/Pillai1.txt">Unique solutions to Pillai's Equation 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, A053289, A074981, A076427, A076438, A076439, A076440, A207079.
%K hard,nonn
%O 1,2
%A _T. D. Noe_, Oct 12 2002