This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A076427 #26 Feb 16 2025 08:32:47 %S A076427 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, %T A076427 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, %U A076427 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 %N A076427 Number of solutions to Pillai's equation a^x - b^y = n, with a>0, b>0, x>1, y>1. %C A076427 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. %D A076427 R. K. Guy, Unsolved Problems in Number Theory, D9. %D A076427 T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986. %H A076427 T. D. Noe, <a href="http://www.sspectra.com/Pillai.txt">Solutions to Pillai's Equation for n<=100</a> %H A076427 Dorin Andrica and Ovidiu Bagdasar, <a href="https://doi.org/10.1007/s11139-021-00418-7">On k-partitions of multisets with equal sums</a>, The Ramanujan J. (2021) Vol. 55, 421-435. %H A076427 Dorin Andrica and George C. Ţurkaş, <a href="https://doi.org/10.24193/subbmath.2019.3.06">An elliptic Diophantine equation from the study of partitions</a>, Stud. Univ. Babeş-Bolyai Math. (2019) Vol. 64, No. 3, 349-356. %H A076427 M. A. Bennett, <a href="http://www.math.ubc.ca/~bennett/B-CJM-Pillai.pdf">On some exponential equations of S. S. Pillai</a>, Canad. J. Math. 53 (2001), 897-922. %H A076427 Dana Mackenzie, <a href="http://math.colgate.edu/~integers/s33/s33.Abstract.html">2184: An absurd (and adsurd) tale</a>, Integers (2018) 18, Article #A33. %H A076427 Roswitha Rissner and Daniel Windisch, <a href="https://arxiv.org/abs/2009.02322">Absolute irreducibility of the binomial polynomials</a>, arXiv:2009.02322 [math.AC], 2020. %H A076427 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PillaisConjecture.html">Pillai's Conjecture</a> %e A076427 a(4)=3 because there are 3 solutions: 4 = 2^3 - 2^2 = 6^2 - 2^5 = 5^3 - 11^2. %Y A076427 Cf. A189117, A001597, A074981. %K A076427 hard,nonn %O A076427 1,3 %A A076427 _T. D. Noe_, Oct 11 2002