

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
Dorin Andrica, George C. Ţurkaş, An elliptic Diophantine equation from the study of partitions, Stud. Univ. BabeşBolyai Math. (2019) Vol. 64, No. 3, 349356.
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.
Roswitha Rissner, Daniel Windisch, Absolute irreducibility of the binomial polynomials, arXiv:2009.02322 [math.AC], 2020.
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: A339451 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



