%I A074981
%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,
%U A074981 310,314,322,326,330,358,374,378,390,394,398,402,410,418,422,426
%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 n such that there is no solution to Pillai's equation. - T. D. Noe (noe(AT)sspectra.com), 
               Oct 12 2002
%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 P. Ribenboim, Catalan's Conjecture, Amer. Math. Monthly, Vol. 103(7) 
               Aug-Sept 1996, pp. 529-538.
%D A074981 T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge 
               University Press, 1986.
%H A074981 A. Baker, <a href="http://www.ams.org/bull/pre-1996-data/199501/199501012.html">
               Review of "Catalan's conjecture" by P. Ribenboim</a>
%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 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/
               page.php?sort=CatalansProblem">Catalan's Problem</a>
%H A074981 MathWorld, <a href="http://mathworld.wolfram.com/news/2002-05-05/catalan/
               ">Catalan Conjecture</a>
%H A074981 T. Metsankyla, <a href="http://www.ams.org/bull/2004-41-01/S0273-0979-03-00993-5/
               home.html">Catalan's Conjecture : Another old Diophantine problem 
               solved</a>
%H A074981 M. Mischler, <a href="http://fr.arXiv.org/abs/math.NT/0502350">La conjecture 
               de Catalan racontee a un ami qui a le temps</a>
%H A074981 Alf van der Poorten, <a href="a023057.txt">Remarks on the sequence of 
               'perfect' powers</a>
%H A074981 Wikipedia, <a href="http://www.wikipedia.org/wiki/Catalan%27s_conjecture">
               Catalan's conjecture</a>
%H A074981 G. Villemin's Almanac of Numbers, <a href="http://perso.wanadoo.fr/yoda.guillaume/
               Nb30a50/CataConj.htm">Conjecture de Catalan</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, ...
%e A074981 342 = 7^3 - 1^2, ...
%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 Cf. A074980, A069586, A023057, A066510, A075788-A075791.
%Y A074981 Sequence in context: A024932 A078836 A142875 this_sequence A066510 A036387 
               A053560
%Y A074981 Adjacent sequences: A074978 A074979 A074980 this_sequence A074982 A074983 
               A074984
%K A074981 nonn,hard
%O A074981 1,1
%A A074981 Zak Seidov (zakseidov(AT)yahoo.com), Oct 07 2002
%E A074981 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.