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A074981
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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.
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22
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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
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OFFSET
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1,1
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COMMENTS
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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
Though this sequence has been generated checking differences of two powers below 10^19, the sequence can be found by the Mma program below using only the first 5000 terms of A001597 and checking differences. - Frank M Jackson, May 20 2017
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REFERENCES
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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.
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LINKS
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Table of n, a(n) for n=1..52.
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, 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
Alf van der Poorten, Remarks on the sequence of 'perfect' powers
P. Ribenboim, Catalan's Conjecture, Séminaire de Philosophie et Mathématiques, 6 (1994), p. 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, Catalan Conjecture
Eric Weisstein's World of Mathematics, Pillai's Conjecture
Wikipedia, Catalan's conjecture
Wikipedia, Hall's conjecture
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EXAMPLE
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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, ...
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MATHEMATICA
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nextPerfectPower[n_] := If[n==1, 4, Min@Table[(Floor[n^(1/k)]+1)^k, {k, 2, 1+Floor@Log2@n}]]; lst=NestList[nextPerfectPower, 1, 5000]; lst1=Union@Flatten@ Table[lst[[x]]-lst[[y]], {x, 1, Length[lst]}, {y, x-1}]; Select[Range[500], !MemberQ[lst1, #] &] (* Frank M Jackson, May 20 2017 using Robert G. Wilson v program A001597 *)
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CROSSREFS
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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.
Cf. A001597, A074980, A069586, A023057, A066510, A075788-A075791, A053289, A074981, A076438, A207079, A219551.
Sequence in context: A225972 A078836 A142875 * A066510 A279730 A269717
Adjacent sequences: A074978 A074979 A074980 * A074982 A074983 A074984
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KEYWORD
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nonn,hard
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AUTHOR
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Zak Seidov, Oct 07 2002
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EXTENSIONS
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Corrected by Don Reble and Jud McCranie, Oct 08 2002. Corrections were also sent in by Neil Fernandez, David W. Wilson, and Reinhard Zumkeller.
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STATUS
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approved
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