A236211 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.

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

Offset: 1

Author

Jonathan Sondow, Jan 23 2014

Keywords

• nonn

Comments

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.

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.

Examples

			3 - 2 = 3^2 - 2^3 = 1.
2^3 - 5 = 2^7 - 5^3 = 3.
6 - 2 = 6^2 - 2^5 = 4.
2^3 - 3 = 2^5 - 3^3 = 5.
15 - 6 = 15^2 - 6^3 = 9.
13 - 3 = 13^3 - 3^7 = 10.
2^4 - 3 = 2^8 - 3^5 = 13.
91 - 2 = 91^2 - 2^13 = 89.
280 - 5 = 280^2 - 5^7 = 275.
6^4 - 3^4 = 6^5 - 3^8 = 1215.
4930 - 30 = 4930^2 - 30^5 = 4900.
		

References

• R. K. Guy, Unsolved Problems in Number Theory, D9.
• T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986.

Links

• M. A. Bennett, On Some Exponential Equations of S. S. Pillai, Canad. J. Math., 53 (2001), 897-922.
• J.-H. Evertse, Review of M. A. Bennett's "On Some Exponential Equations of S. S. Pillai", zbMATH 0984.11014
• OEIS, Entries related to Pillai's equation
• M. Waldschmidt, Open Diophantine problems
• E. Weisstein's MathWorld, Pillai's Conjecture

Crossrefs

Cf. A207079 and the OEIS link.