

A074981


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.


22



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


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 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
The terms shown are not the difference of two powers below 10^27.  Mauro Fiorentini, Jan 03 2020


REFERENCES

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.


LINKS



EXAMPLE

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, ...


CROSSREFS

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, A075788A075791, A053289, A074981, A076438, A207079, A219551.


KEYWORD

nonn,hard


AUTHOR



EXTENSIONS



STATUS

approved



