Generating a specific list of intsgers

[Musatov]

On Sunday, August 26, 2018 at 2:14:29 PM UTC-5, Oscar Benjamin wrote:
> On Sat, 25 Aug 2018 at 20:27, Musatov <tomusatov at gmail.com> wrote:
> >
> > On Saturday, August 25, 2018 at 2:18:09 PM UTC-5, Musatov wrote:
> > > On Saturday, August 25, 2018 at 1:52:17 PM UTC-5, Oscar Benjamin wrote:
> > > > > > >
> > > > > > >> On Fri, 24 Aug 2018 14:40:00 -0700, tomusatov wrote:
> > > > > > >>
> > > > > > >>> I am looking for a program able to output a set of integers meeting the
> > > > > > >>> following requirement:
> > > > > > >>>
> > > > > > >>> a(n) is the minimum k > 0 such that n*2^k - 3 is prime, or 0 if no such
> > > > > > >>> k exists
> > > > > > >>>
> > > > > > >>> Could anyone get me started? (I am an amateur)
> > > > > > >>
> > > > > > >> That's more a maths question than a programming question. Find out how to
> > > > > > >> tackle it mathematically, and then we can code it.
> > > >
> > > > Looks like it's zero for any multiple of 3 (apart from 3 itself). This
> > > > makes sense since if n is a equal to b*3 for some integer b then
> > > >     n*2^k - 3 = b*3*2^k - 3 = (b*2^k - 1)*3
> > > > which can only be prime if
> > > >     b*2^k - 1 = 1
> > > > which can only be true if b=1 (since k>0) implying that n=3. So for
> > > > any *other* multiple of 3 you must necessarily have a(n) = 0.
> > > >
> > > > The above means that you can handle all multiples of 3 but how do you
> > > > know that you won't hit an infinite loop when n is not a multiple of
> > > > 3?
> >
> > Rather, I should say one such n is 72726958979572419805016319140106929109473069209 (which is not divisible by 3)
> 
> Fair enough. So finding a(n) when a(n)!=0 is straight-forward (simply
> loop through testing k=1,2...) but the issue is determining for any
> given n whether a(n)=0 i.e. that there does not exist k such that
> n*2^k-3 is prime.
> 
> Perhaps if you explain how you know that
>    a(72726958979572419805016319140106929109473069209) = 0
> then that would suggest a way to code it.
> 
> --
> Oscar
Oscar, I simply asked someone and they provided me the number. I know they often use Maple, but I was interested in Python.

He also said some of the n are prime by Dirichlet's theorem. One is 8236368172492875810638652252525796530412199592269.