

A076427


Number of solutions to Pillai's equation a^x  b^y = n, with a>0, b>0, x>1, y>1.


5



1, 1, 2, 3, 2, 0, 5, 3, 4, 1, 4, 2, 3, 0, 3, 3, 7, 3, 5, 2, 2, 2, 4, 5, 2, 3, 3, 7, 1, 1, 2, 4, 2, 0, 3, 2, 3, 1, 4, 4, 3, 0, 1, 3, 4, 1, 6, 4, 3, 0, 2, 1, 2, 2, 3, 4, 3, 0, 1, 4, 2, 0, 4, 4, 4, 0, 2, 5, 2, 0, 4, 4, 6, 2, 3, 3, 2, 0, 4, 4, 4, 0, 2, 2, 2, 0, 3, 3, 6, 0, 3, 4, 4, 2, 4, 5, 3, 2, 4, 10
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OFFSET

1,3


COMMENTS

This is the classic Diophantine equation of S. S. Pillai, who conjectured that there are only a finite number of solutions for each n. A generalization of Catalan's conjecture that a^xb^y=1 has only one solution. For n <=100, a total of 274 solutions were found for perfect powers less than 10^12. No additional solutions were found for perfect powers < 10^18.


REFERENCES

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


LINKS

Table of n, a(n) for n=1..100.
T. D. Noe, Solutions to Pillai's Equation for n<=100
M. A. Bennett, On some exponential equations of S. S. Pillai, Canad. J. Math. 53 (2001), 897922.
Dana Mackenzie, 2184: An absurd (and adsurd) tale, Integers (2018) 18, Article #A33.
Eric Weisstein's World of Mathematics, Pillai's Conjecture


EXAMPLE

a(4)=3 because there are 3 solutions: 4 = 2^3  2^2 = 6^2  2^5 = 5^3  11^2.


CROSSREFS

Cf. A189117, A001597, A074981.
Sequence in context: A286297 A111182 A178142 * A284152 A011024 A105855
Adjacent sequences: A076424 A076425 A076426 * A076428 A076429 A076430


KEYWORD

hard,nonn


AUTHOR

T. D. Noe, Oct 11 2002


STATUS

approved



