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A066510 Conjectured list of positive numbers which are not of the form r^i-s^j, where r,s,i,j are integers with i>1, j>1.

%I A066510 #20 Jan 11 2025 04:08:34
%S A066510 6,14,34,42,58,62,66,70,78,86,90,102,110,114,130,158,178,182,202,210,
%T A066510 230,238,254,258,266,274,278,302,306,310,314,322,326,330,358,374,378,
%U A066510 390,394,398,402,410,418,422,426,430,434,438,446,450,454
%N A066510 Conjectured list of positive numbers which are not of the form r^i-s^j, where r,s,i,j are integers with i>1, j>1.
%C A066510 This is a famous hard problem and the terms shown are only conjectured values.
%C A066510 The terms shown are not the difference of two powers below 10^19. - _Don Reble_
%C A066510 One can immediately represent the odd numbers and the multiples of four as differences of two squares. - _Don Reble_
%C A066510 The terms shown are not the difference of two powers below 10^27. - _Mauro Fiorentini_, Jan 08 2020
%D A066510 R. K. Guy, Unsolved Problems in Number Theory, Sections D9 and B19.
%H A066510 Mauro Fiorentini, <a href="/A066510/b066510.txt">Table of n, a(n) for n = 1..119</a>
%H A066510 Alf van der Poorten, <a href="/A023057/a023057.txt">Remarks on the sequence of 'perfect' powers</a>.
%e A066510 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 A066510 50 = 7^2 - -1^3, 82 = 9^2 - -1^3, 226 = 15^2 - -1^3, 246 = 11^2 - -5^3, 290 = 17^2 - -1^3, ... [Typos corrected by _Gerry Myerson_, May 14 2008]
%Y A066510 Cf. A074980, A023057.
%Y A066510 For sequence with similar definition, but restricted to positive values of r and s, see A074981.
%K A066510 nonn,hard
%O A066510 1,1
%A A066510 _Don Reble_, Oct 12 2002