# not of form m^p-n^q

N. J. A. Sloane njas at research.att.com
Thu Oct 10 00:10:33 CEST 2002

```There has been a lot of discussion about this "sequence",
so I have put a version of it in the database:

%I A074981
%S A074981 6,14,34,42,50,58,62,66,70,78,82,86,90,102,110,114,130,158,178,182,202,
%T A074981 210,226,230,238,246,254,258,266,274,278,290,302,306,310,314,322,326,
%U A074981 330,342,358,374,378,390,394,398,402,410,418,422,426,430,434,438,442
%N A074981 Conjectured list of numbers which are not of form m^p-n^q, where m,n,p,q are integers with m>0, n>0, p>1, q>1.
%C A074981 These are not the difference of two powers below 10^19. - Don Reble.
%e A074981 Examples for numbers 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, ...
%Y A074981 Cf. A074980.
%K A074981 nonn,new
%O A074981 1,1
%A A074981 Zakir F. Seidov (seidovzf at yahoo.com), Oct 07 2002
%E A074981 Corrected by Don Reble (djr at nk.ca), Oct 08 2002

The terms may change, and the %E (Extended) line
will probably change too, when i process all the messages
and postings dealin with this sequence.  But at least
it now has an A-number.

There is a closely related sequence that will also probably
change:

%I A074980
%S A074980 6,10,14,34,42,46,50,58,62,66,70,78,82,86,90,102,110,114,122,130,134,
%T A074980 146,150,158,162,166,178,182,194
%N A074980 Numbers which are not of the form m^p - n^q where p = 2 or 3, q = 2 or 3.
%C A074980 Take d=m^p - n^q, (p,q = 2,3) that is difference between cube/square of any m and cube/square of any n. Presumably the absolute value of d can not take the values 6, 10, 14, 34, 42, ... in the sequence.
%e A074980 First squares and cubes are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 8, 27, 64. Take differences between any two terms. The absolute values of these differences can not be (presumably) equal to numbers in the sequence, e.g. 6, or 10.
%Y A074980 Cf. A074981.
%K A074980 nonn,new
%O A074980 1,1
%A A074980 Zakir F. Seidov (seidovzf at yahoo.com), Oct 07 2002
%E A074980 Probably wrong. Will be modified. - njas

Although I don't think this has received much attention yet.
Maybe someone could check it.

Neil

```