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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.

%I A074981 #112 Feb 16 2025 08:32:47
%S A074981 6,14,34,42,50,58,62,66,70,78,82,86,90,102,110,114,130,134,158,178,
%T A074981 182,202,206,210,226,230,238,246,254,258,266,274,278,290,302,306,310,
%U A074981 314,322,326,330,358,374,378,390,394,398,402,410,418,422,426
%N 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.
%C A074981 This is a famous hard problem and the terms shown are only conjectured values.
%C A074981 The terms shown are not the difference of two powers below 10^19. - _Don Reble_
%C A074981 One can immediately represent all odd numbers and multiples of 4 as differences of two squares. - _Don Reble_
%C A074981 _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.
%C A074981 Numbers n such that there is no solution to Pillai's equation. - _T. D. Noe_, Oct 12 2002
%C A074981 The terms shown are not the difference of two powers below 10^27. - _Mauro Fiorentini_, Jan 03 2020
%D A074981 R. K. Guy, Unsolved Problems in Number Theory, Sections D9 and B19.
%D A074981 P. Ribenboim, Catalan's Conjecture, Academic Press NY 1994.
%D A074981 T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986.
%H A074981 Mauro Fiorentini, <a href="/A074981/b074981.txt">Table of n, a(n) for n = 1..138</a>
%H A074981 A. Baker, <a href="http://dx.doi.org/10.1090/S0273-0979-1995-00559-8">Review of "Catalan's conjecture" by P. Ribenboim</a>, Bull. Amer. Math. Soc. 32 (1995), 110-112.
%H A074981 M. E. Bennett, <a href="http://www.math.ubc.ca/~bennett/paper19.pdf">On Some Exponential Equations Of S. S. Pillai</a>, Canad. J. Math. 53 (2001), 897-922.
%H A074981 Yu. F. Bilu, <a href="http://www.math.u-bordeaux.fr/~yuri/publ/preprs/catal.pdf">Catalan's Conjecture (after Mihailescu)</a>
%H A074981 J. Boéchat and M. Mischler, <a href="http://arxiv.org/abs/math/0502350">La conjecture de Catalan racontée a un ami qui a le temps</a>, arXiv:math/0502350 [math.NT], 2005-2006.
%H A074981 C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=CatalansProblem">Catalan's Problem</a>
%H A074981 T. Metsankyla, <a href="http://dx.doi.org/10.1090/S0273-0979-03-00993-5">Catalan's Conjecture: Another old Diophantine problem solved</a>, Bull. Amer. Math. Soc. 41 (2004), 43-57.
%H A074981 Alf van der Poorten, <a href="/A023057/a023057.txt">Remarks on the sequence of 'perfect' powers</a>
%H A074981 P. Ribenboim, <a href="http://www.numdam.org/item?id=SPHM_1994___6_A1_0">Catalan's Conjecture</a>, Séminaire de Philosophie et Mathématiques, 6 (1994), pp. 1-11.
%H A074981 P. Ribenboim, <a href="http://www.jstor.org/stable/2974663">Catalan's Conjecture</a>, Amer. Math. Monthly, Vol. 103(7) Aug-Sept 1996, pp. 529-538.
%H A074981 Gérard Villemin, <a href="http://villemin.gerard.free.fr/aNombre/TYPDENOM/Catalan/CataConj.htm">Conjecture de Catalan</a> (French)
%H A074981 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/news/2002-05-05/catalan/">Draft Proof of Catalan's Conjecture Circulated</a>
%H A074981 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PillaisConjecture.html">Pillai's Conjecture</a>
%H A074981 Wikipedia, <a href="http://www.wikipedia.org/wiki/Catalan%27s_conjecture">Catalan's conjecture</a>
%H A074981 Wikipedia, <a href="https://www.wikipedia.org/wiki/Hall%27s_conjecture">Hall's conjecture</a>
%e A074981 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, ...
%Y A074981 Subsequence of A016825 (see second comment of _Don Reble_).
%Y A074981 n such that A076427(n) = 0. [Corrected by _Jonathan Sondow_, Apr 14 2014]
%Y A074981 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.
%Y A074981 For sequence with similar definition, but allowing negative powers, see A066510.
%Y A074981 Cf. A001597, A074980, A069586, A023057, A075788-A075791, A053289, A076438, A207079, A219551.
%K A074981 nonn,hard
%O A074981 1,1
%A A074981 _Zak Seidov_, Oct 07 2002
%E A074981 Corrected by _Don Reble_ and _Jud McCranie_, Oct 08 2002. Corrections were also sent in by _Neil Fernandez_, _David W. Wilson_, and _Reinhard Zumkeller_.