A066510 - OEIS

%I #16 May 18 2024 00:50:48

%S 6,14,34,42,58,62,66,70,78,86,90,102,110,114,130,158,178,182,202,210,

%T 230,238,254,258,266,274,278,302,306,310,314,322,326,330,358,374,378,

%U 390,394,398,402,410,418,422,426,430,434,438,446,450,454

%N 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 This is a famous hard problem and the terms shown are only conjectured values.

%C The terms shown are not the difference of two powers below 10^19. - _Don Reble_

%C One can immediately represent the odd numbers and the multiples of four as differences of two squares. - _Don Reble_

%C The terms shown are not the difference of two powers below 10^27. - _Mauro Fiorentini_, Jan 08 2020

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

%H Mauro Fiorentini, <a href="/A066510/b066510.txt">Table of n, a(n) for n = 1..119</a>

%H Alf van der Poorten, <a href="/A023057/a023057.txt">Remarks on the sequence of 'perfect' powers</a>.

%e 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 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 Cf. A074980, A023057.

%K nonn,hard

%O 1,1

%A _Don Reble_, Oct 12 2002