Re: Q: discrete logarithms and modular powerseries [Iktc'tpw-msowlttmwibd]

Am 13.05.05 03:09 schrieb Gerry Myerson:

> What makes things difficult is that the polynomial you need depends,
> in no simple way, on the base & modulus. Also, its degree will be,
> roughly, the modulus; in cases of practical interest, a dense
> polynomial of that degree will be useless.
>
Hmm, that's an argument. The length of the modular periodicity
of powers of base b is some integer part of the value of the
phi-function of n, (unfortunately) dependend on both parameters (n,b).
For b=2 there are some additional properties, for instance
dependend on the modulus of n base 8, which defines the lengthes
*for primes* even (with factor 2 resp 4 for n=8i+-3) and odd
for primes of the structure 8i-1, and mixed for primes of
the structure 8i+1.
All such lengthes and the prime-factorization of 2^n-1 seem
to be identifyable connected, such that if the primefactors
of 2^n-1 have certain lenghtes, then -in a sense- these lengthes
"are no more available" for other primes (don't know how to express
this better currently, sorry)

One reason, why I'm on this topic, is for instance the extremely simple
function for numbers of the form 2^k-1 (which simply have the length k)
and the lengthes for prime-powers (coprime to th base) p^a,
which are a factor of the phi-function, many of them are just

length(p^a,base) = phi(p^a)/m = (p-1)/m * p^(a-1).

(where for the phi-function m=1, and m dependend on (p,b))

With an observation like that one can then access the question
of sequential coprime powers like 2^a - 1 =?= p^b
and things like that.

It's not very professional things, sorry.

>
> I wish people would preface their posts with something like,
> "I know this can't possibly work - maybe someone would like
> to tell me where it breaks down."
>
Well - I could take this as a sig for s.m.... or are you suggesting
to put that sig even on top of such postings? As a shorthand, I
could add a new acronym "[Iktc'tpw-msowlttmwibd]" to the subject? :-)))

Done -

Gottfried Helms
.