1,1

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.

n such that there is no solution to Pillai's equation. - T. D. Noe (noe(AT)sspectra.com), Oct 12 2002

R. K. Guy, Unsolved Problems in Number Theory, Sections D9 and B19.

P. Ribenboim, Catalan's Conjecture, Academic Press NY 1994.

P. Ribenboim, Catalan's Conjecture, Amer. Math. Monthly, Vol. 103(7) Aug-Sept 1996, pp. 529-538.

T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986.

A. Baker, Review of "Catalan's conjecture" by P. Ribenboim

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)

C. K. Caldwell, The Prime Glossary, Catalan's Problem

MathWorld, Catalan Conjecture

T. Metsankyla, Catalan's Conjecture : Another old Diophantine problem solved

M. Mischler, La conjecture de Catalan racontee a un ami qui a le temps

Alf van der Poorten, Remarks on the sequence of 'perfect' powers

Wikipedia, Catalan's conjecture

G. Villemin's Almanac of Numbers, Conjecture de Catalan

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, ...

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.

Cf. A074980, A069586, A023057, A066510, A075788-A075791.

Sequence in context: A024932 A078836 A142875 this_sequence A066510 A036387 A053560

Adjacent sequences: A074978 A074979 A074980 this_sequence A074982 A074983 A074984

nonn,hard

Zak Seidov (zakseidov(AT)yahoo.com), Oct 07 2002

Corrected by Don Reble (djr(AT)nk.ca) and Jud McCranie, Oct 08 2002. Corrections were also sent in by N. Fernandez, David W. Wilson, reinhard.zumkeller(AT)gmail.com.