A236211 - OEIS

%I #6 Jan 24 2014 01:11:07

%S 1,3,4,5,9,10,13,89,275,1215,4900

%N Numbers c > 0 for which there exist integers a > 1 and b > 1 such that the equation a^x - b^y = c has two solutions in positive integers x, y.

%C Bennett proved that if a, b, c are nonzero integers with a > 1 and b > 1, then the equation a^x - b^y = c has at most two solutions in positive integers x and y.

%C Bennett conjectured that if a, b, c are positive integers with a > 1 and b > 1, then the equation a^x - b^y = c has at most one solution in positive integers x and y, except for the triples (a,b,c) = (3,2,1), (2,5,3), (6,2,4), (2,3,5), (15,6,9), (13,3,10), (2,3,13), (91,2,89), (280,5,275), (6,3,1215), (4930,30,4900). If this is true, then the present sequence is complete.

%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. A. Bennett, <a href="http://cms.math.ca/cjm/v53/bennett1355.pdf">On Some Exponential Equations of S. S. Pillai</a>, Canad. J. Math., 53 (2001), 897-922.

%H J.-H. Evertse, <a href="http://zbmath.org/?q=an:0984.11014">Review of M. A. Bennett's "On Some Exponential Equations of S. S. Pillai"</a>, zbMATH 0984.11014

%H OEIS, <a href="https://oeis.org/search?q=%22pillai%27s+equation">Entries related to Pillai's equation</a>

%H M. Waldschmidt, <a href="http://arXiv.org/abs/math.NT/0312440">Open Diophantine problems</a>

%H E. Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/PillaisConjecture.html">Pillai's Conjecture</a>

%e 3 - 2 = 3^2 - 2^3 = 1.

%e 2^3 - 5 = 2^7 - 5^3 = 3.

%e 6 - 2 = 6^2 - 2^5 = 4.

%e 2^3 - 3 = 2^5 - 3^3 = 5.

%e 15 - 6 = 15^2 - 6^3 = 9.

%e 13 - 3 = 13^3 - 3^7 = 10.

%e 2^4 - 3 = 2^8 - 3^5 = 13.

%e 91 - 2 = 91^2 - 2^13 = 89.

%e 280 - 5 = 280^2 - 5^7 = 275.

%e 6^4 - 3^4 = 6^5 - 3^8 = 1215.

%e 4930 - 30 = 4930^2 - 30^5 = 4900.

%Y Cf. A207079 and the OEIS link.

%K nonn

%O 1,2

%A _Jonathan Sondow_, Jan 23 2014