A074981 Conjectured list of positive numbers which are not of the form r^i - s^j, where r,s,i,j are integers with r>0, s>0, i>1, j>1.

6, 14, 34, 42, 50, 58, 62, 66, 70, 78, 82, 86, 90, 102, 110, 114, 130, 134, 158, 178, 182, 202, 206, 210, 226, 230, 238, 246, 254, 258, 266, 274, 278, 290, 302, 306, 310, 314, 322, 326, 330, 358, 374, 378, 390, 394, 398, 402, 410, 418, 422, 426

Offset: 1

Author

Zak Seidov, Oct 07 2002

Keywords

• nonn
• hard

Comments

This is a famous hard problem and the terms shown are only conjectured values.

The terms shown are not the difference of two powers below 10^19. - Don Reble

One can immediately represent all odd numbers and multiples of 4 as differences of two squares. - Don Reble

_Ed Pegg Jr_ remarks (Oct 07 2002) that the techniques of Preda Mihailescu (see MathWorld link) might make it possible to prove that 6, 14, ... are indeed members of this sequence.

Numbers n such that there is no solution to Pillai's equation. - T. D. Noe, Oct 12 2002

The terms shown are not the difference of two powers below 10^27. - Mauro Fiorentini, Jan 03 2020

Examples

			Examples showing that certain numbers are not in the sequence: 10 = 13^3 - 3^7, 22 = 7^2 - 3^3, 29 = 15^2 - 14^2, 31 = 2^5 - 1, 52 = 14^2 - 12^2, 54 = 3^4 - 3^3, 60 = 2^6 - 2^2, 68 = 10^2 - 2^5, 72 = 3^4 - 3^2, 76 = 5^3 - 7^2, 84 = 10^2 - 2^4, ... 342 = 7^3 - 1^2, ...
		

References

• R. K. Guy, Unsolved Problems in Number Theory, Sections D9 and B19.
• P. Ribenboim, Catalan's Conjecture, Academic Press NY 1994.
• T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986.

Links

• Mauro Fiorentini, Table of n, a(n) for n = 1..138
• A. Baker, Review of "Catalan's conjecture" by P. Ribenboim, Bull. Amer. Math. Soc. 32 (1995), 110-112.
• M. E. Bennett, On Some Exponential Equations Of S. S. Pillai, Canad. J. Math. 53 (2001), 897-922.
• Yu. F. Bilu, Catalan's Conjecture (after Mihailescu)
• J. Boéchat and M. Mischler, La conjecture de Catalan racontée a un ami qui a le temps, arXiv:math/0502350 [math.NT], 2005-2006.
• C. K. Caldwell, The Prime Glossary, Catalan's Problem
• T. Metsankyla, Catalan's Conjecture: Another old Diophantine problem solved, Bull. Amer. Math. Soc. 41 (2004), 43-57.
• Alf van der Poorten, Remarks on the sequence of 'perfect' powers
• P. Ribenboim, Catalan's Conjecture, Séminaire de Philosophie et Mathématiques, 6 (1994), pp. 1-11.
• P. Ribenboim, Catalan's Conjecture, Amer. Math. Monthly, Vol. 103(7) Aug-Sept 1996, pp. 529-538.
• Gérard Villemin, Conjecture de Catalan (French)
• Eric Weisstein's World of Mathematics, Draft Proof of Catalan's Conjecture Circulated
• Eric Weisstein's World of Mathematics, Pillai's Conjecture
• Wikipedia, Catalan's conjecture
• Wikipedia, Hall's conjecture

Crossrefs

Subsequence of A016825 (see second comment of Don Reble).

n such that A076427(n) = 0. [Corrected by Jonathan Sondow, Apr 14 2014]

For a count of the representations of a number as the difference of two perfect powers, see A076427. The numbers that appear to have unique representations are listed in A076438.

For sequence with similar definition, but allowing negative powers, see A066510.

Cf. A001597, A074980, A069586, A023057, A075788-A075791, A053289, A076438, A207079, A219551.

Extensions

Corrected by Don Reble and Jud McCranie, Oct 08 2002. Corrections were also sent in by Neil Fernandez, David W. Wilson, and Reinhard Zumkeller.