A076427 Number of solutions to Pillai's equation a^x - b^y = n, with a>0, b>0, x>1, y>1.

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

Offset: 1

Author

T. D. Noe, Oct 11 2002

Keywords

• hard
• nonn

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^x-b^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.

Examples

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

References

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

Links

• T. D. Noe, Solutions to Pillai's Equation for n<=100
• Dorin Andrica and Ovidiu Bagdasar, On k-partitions of multisets with equal sums, The Ramanujan J. (2021) Vol. 55, 421-435.
• Dorin Andrica and George C. Ţurkaş, An elliptic Diophantine equation from the study of partitions, Stud. Univ. Babeş-Bolyai Math. (2019) Vol. 64, No. 3, 349-356.
• M. A. Bennett, On some exponential equations of S. S. Pillai, Canad. J. Math. 53 (2001), 897-922.
• Dana Mackenzie, 2184: An absurd (and adsurd) tale, Integers (2018) 18, Article #A33.
• Roswitha Rissner and Daniel Windisch, Absolute irreducibility of the binomial polynomials, arXiv:2009.02322 [math.AC], 2020.
• Eric Weisstein's World of Mathematics, Pillai's Conjecture

Crossrefs

Cf. A189117, A001597, A074981.