>
> Given the discussion recently about double checking exponents, I think
> it would be worthwhile if someone could post a summary of what if
> any different starting numbers are possible for LL tests and any papers
> that deal with this.
This has been discussed before. Searching for "starting values"
in the Mersenne mailing list archive I find
http://www2.netdoor.com/~acurry/mersenne/archive3/0679.html
I would also suggest reading the articles in Luke's Mersenne
Bibliography http://www.scruznet.com/~luke/biblio.htm
The best paper on this subject in my opinion is Lehmer, D.H. "An
Extended Theory of Lucas' Functions." Annals of Math., v. 31, p.419-448.,
1930 esp. Section 5 (Tests for Primality).
I'll restate the main points. For the recurrence S{k}=S{k-1}^2-2 mod N
there are an infinite number of starting values,
S{1}=4,52,724,..., u{n}, ..., where u{n}=14*u{n-1} - u{n-2} and
S{1}=10,970,95050,..., u{n}, ..., where u{n}=98*u{n-1} - u{n-2}.
There are also starting values that are dependent on p. In fact
2^(p-2) starting values. For example to test 2^5-1 you can use one
of the eight starting values 4, 9, 10, 11, 20, 21, 22, or 27.
Unfortunately, two LL-tests using two different starting values
will have different residues.
