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.

6, 14, 34, 42, 58, 62, 66, 70, 78, 86, 90, 102, 110, 114, 130, 158, 178, 182, 202, 210, 230, 238, 254, 258, 266, 274, 278, 302, 306, 310, 314, 322, 326, 330, 358, 374, 378, 390, 394, 398, 402, 410, 418, 422, 426, 430, 434, 438, 446, 450, 454

Offset: 1

Author

Don Reble, Oct 12 2002

Keywords

• nonn
• hard

Comments

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 the odd numbers and the multiples of four as differences of two squares. - Don Reble

The terms shown are not the difference of two powers below 10^27. - Mauro Fiorentini, Jan 08 2020

Examples

			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, ...
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]
		

References

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

Links

• Mauro Fiorentini, Table of n, a(n) for n = 1..119
• Alf van der Poorten, Remarks on the sequence of 'perfect' powers.

Crossrefs

Cf. A074980, A023057.

For sequence with similar definition, but restricted to positive values of r and s, see A074981.