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# A question about prime divisors of Mersenne number $M_n= 2^n-1$ when $n$ is odd

Is this true that a prime divisor of a Mersenne number $M_n = 2^n-1$ when $n$ is odd, cannot be a Proth prime (i.e. a prime number of the form $2^mk+1$, where $k<2^m$)?

If yes, how is it demonstrated?

thank you!

because, if I am not wrong, for any prime $p>3$ ($= r2^v +1$, where $r$ is odd and $v>1$),
either there exists a unique $j < v-1$ so that $p$ is a divisor of $2^J+1$, where $J=r2^j$,
or $p$ is divisor of $2^r-1$.

In the latter case, whenever I have checked so far, if I write $p$ as $k2^m+1$, ($k$ odd) $k$ is larger than $2^m$. Is this always like that?

In the former case, most of the times $j$ is $v-2$ or a slightly smaller. In particular, so far I have checked, it is rather rare that $v$ and $j$ differ by more than say $5$, like for instance when $p=65537(j=4,v=16)$, or $p=59393(j=2,v=11)$, or $p=25601 (j=3,v=10)$, or $p=2113 (j=1,v=6)$ or $p=6529 (j=0,v=7)$... I wonder how many such prime numbers exist ?

• Why do you suspect this should be true? – Greg Martin Feb 19 '14 at 18:39

Interesting conjecture! It holds for quite some time...

... before failing at $n=225$. The corresponding Proth prime factor is $p=115201=225\times 2^9+1$. It's not too difficult to see that any odd multiple of $n$ would be a counter-example too, since the corresponding Mersenne number would be divisible by $p$ as well.

The next two non-trivial cases seem to be:

• $n=6281$ and $p=51453953=6281\times 2^{13}+1$
• $n=7695$ and $p=126074881=7695\times 2^{14}+1$

Interestingly, those are all the counter-examples I managed to find even after searching for quite some more time. It's not too surprising, though, since Proth primes are relatively rare.

• thank you for the counterexamples! I did not go that far... – René Gy Feb 20 '14 at 1:16
• it seems that $m$ is hardly much bigger than than the exponent in the biggest power of $2$ which is smaller than $n$... but again this should checked for the very big $n$... – René Gy Feb 20 '14 at 7:09


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## Re: How to generate a .eps file ?

 From: Daniel Heiserer Subject: Re: How to generate a .eps file ? Date: Wed, 27 Jan 1999 10:25:23 +0100

```Daniel Tourde wrote:
>
> Hi !
>
> I use Octave to make computation and to display the results with Gnuplot
> (Nothing extraordinary until now...). Now I would like to put the graphs
> in my tex report. To do so, I need to generate a .eps file (encapsulated
> postscript).
>
> Here is how I proceed :
>
> clearplot
> gset term postscript landscape color

Well as for good tools there are always more than one way to go (perl
philosophy) :-).
1st)
"gset term postscript eps color"
options available via gnuplot and "help set;help term help postscript"
2nd)
tex or let's say latex, what most people use has many import features
for
eps files. With some you can rotate and zoom the plot easily.
"\usepackage{amsmath,latexsym,german,epsfig,graphics,color,umlaute,pifont}
\epsfig{file=bilder/myeps.eps,angle=90,width=80mm}
"
There are really many features including eps-files in your report.

> gset output "/home/lorient/ffa/ase/octave/ase2/data/misc/bessj.eps"
> title ("Bessel functions of the first kind, Jn(x)");
> xlabel ("x")
> ylabel ("y=Jn(x)")
>
> # Postscript ploting
> gplot fig0 title "n=0"
>
> I obtain a postscipt file but unfortunately it's in the wrong position
> (portrait instead of landscape) as I want it to be. I may be wrong in
> the commands I give.
>
> Does anybody know the right way to do it ?
>
> Thanks in advance for any information.
>
>                                 Daniel
>
> --
> ***********************************************************************
> Daniel TOURDE                                   E-mail : address@hidden
> The Aeronautical Research Institute of Sweden   Tel : +46 8 55 54 93 44
> P.O. Box 11021 S-161 11 BROMMA, Sweden          Fax : +46 8    25 34 81
> ***********************************************************************

--
Mit freundlichen Gruessen
Daniel Heiserer
-----
--------------------------------------------------------------
Dipl.-Phys. Daniel Heiserer, BMW AG, Knorrstrasse 147, 80788 Muenchen
Abteilung EK-20
Tel.: 089-382-21187, Fax.: 089-382-42820

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