10=m^p-n^q?
Richard Guy
rkg at cpsc.ucalgary.ca
Tue Oct 8 00:47:44 CEST 2002
Now in B16, I find that Golomb conjectured
that 6 is not the difference of two powererful
numbers (probably not the first to do so)
and Narkiewicz replied with 6 = 5^4*7^3- 463^2
aand noted that there were infinitely many such
representations. Presumably it's still open
whether there are two pure powers that will do
the trick. R.
On Mon, 7 Oct 2002, Richard Guy wrote:
> There's an enormous literature. See D9 in UPINT
> for a sample. There's a connexion with the
> ABC-conjecture -- B19. I'd like to know if you
> fill in any of the gaps below, and would also
> like to see an extended list of gaps. This
> poses a curious question: what about the sequence
>
> 6, 14, 34, 42, 50, 58, 62, 66, 70, 78, ...
>
> whose members are liable to evaporate down the
> years! There are other examples, e.g.,
> numbers not known to be the sum of 3 cubes.
>
> R.
>
> On Mon, 7 Oct 2002, Don Reble wrote:
>
> > > i've just speculated that somel integers can't be represented as
> > > "difference of two full powers >=2"
> >
> > One can immediately represent the odd numbers and the multiples of four,
> > as differences of two squares. Of the rest,
> >
> > 2 = 3^3 - 5^2
> > 6
> > 10 = 13^3 - 3^7 (Thanks, Neil)
> > 14
> > 18 = 3^3 - 3^2
> > 22 = 7^2 - 3^3
> > 26 = 3^3 - 1^2
> > 30 = 83^2 - 19^3
> > 34
> > 38 = 37^2 - 11^3
> > 42
> > 46 = 17^2 - 3^5
> > 50
> > 54 = 3^4 - 3^3
> > ...
> >
> > I haven't yet found reps for 6, 14, 34, 42, 50, 58, 62, 66, 70, 78, ...
> > Zak, do you have one for 6?
> >
> > --
> > Don Reble djr at nk.ca
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