OFFSET
1,2
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.
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
M. Waldschmidt, Open Diophantine problems
E. Weisstein's MathWorld, Pillai's Conjecture
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Jan 23 2014
STATUS
approved