Quote:
Originally Posted by ZFR

E is not just the product of the primes < B1, it's the product of maximum integral powers of each prime fitting < B1 individually, called powersmooth.
The exponentiation
a^{E*2*p} is done mod Mp. So smaller
a doesn't really save run time, or operand size after relatively few operations. In practice the exponentiation is done with a full length fft transform from the start, so it saves no run time to use a smaller base.
https://en.wikipedia.org/wiki/Pollar...92_1_algorithm
As to why not 2 instead of 3, there's a smallnumbers example early in
https://magazine.odroid.com/article/...ticalhistory/ Later on, this same source includes, also in the context of primality testing, rather than P1 factoring, the following:
Code:
In the more general context of testing numbers of arbitrary size, however,
it is important to realize that there are certain classes of numbers, all
of which are Fermat base2 pseudoprimes, irrespective of whether they are
prime or composite. The two bestknown such classes are, firstly, the
Mersenne numbers M(p) = 2^{p}−1 (for which we restrict the definition to prime
exponents since that is required for a number of this form to have a chance
of being prime); for example, 2^{11}−1 passes the test even though it factors as
23 × 89. The second class of such numbers is the Fermat numbers