[QUOTE=maxal]First off, I've got slightly different result for F(43) running your program here:
[code]? F(43)
q: 43 A4188889-28 A4188889-56[/code][/QUOTE] I wrote a program A that displays each factor only once. Then I wrote a simplified version F in order to provide it on the forum. And they seem to differ in some cases, like for 43. So, I've provided result of A(43) instead of F(43). Sorry.
[QUOTE]
Regarding the algorithm: I think there is no much sense in computing gcd like A=gcd(S1-S1_,S2-S2_) and then checking if A is a dividor of M. It is no better than computing gcd(S1-S1_,M) and gcd(S2-S2_,M) and checking if either is greater than 1. But the latter approach allows to process different starting values independently. [/QUOTE]My idea was that mixing values computed from different initial values would lead to more chances to find a factor. Your program shows that the basic Rho Pollard does provide nearly the same factors, but not exactly.[QUOTE]And I am not convinced that using several starting values helps much.[/QUOTE]Your example with 43 shows that 2/3 sometimes provides different factors.

So, do you agree that the conclusion is that this method could find some interesting factors, but with a higher cost than ways already used by prime95 ?
(up to now, I've found NO interesting big factor with this method and Pari ...)
Tony

[QUOTE=T.Rex]My idea was that mixing values computed from different initial values would lead to more chances to find a factor.[/quote]
Mixing values does not help at all. Because of the following trivial observation:
If gcd(S1-S1_,S2-S2_) gives you a factor of M, then the same factor is contained (maybe with others) in both gcd(S1-S1_,M) and gcd(S2-S2_,M).

[QUOTE=T.Rex]Your example with 43 shows that 2/3 sometimes provides different factors.[/quote] Yes, but the amount of work is doubled/tripled/etc. with the number of different starting values.

[QUOTE=T.Rex]So, do you agree that the conclusion is that this method could find some interesting factors, but with a higher cost than ways already used by prime95 ?[/QUOTE]
Maybe.