OFFSET
1,2
COMMENTS
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.
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. E. Bennett, On Some Exponential Equations Of S. S. Pillai, Canad. J. Math. 53 (2001), 897-922.
Eric Weisstein's World of Mathematics, Pillai's Conjecture.
CROSSREFS
KEYWORD
hard,nonn
AUTHOR
T. D. Noe, Oct 12 2002
STATUS
approved