For the first stage, the P-1 algorithm uses the highest power of each prime less than or equal to B1 when computing the product of the primes. However, this means we'll miss factors where
k has a repeated factor greater than sqrt(B1).
102646067 and
105144859 are two examples with such factors. P-1 cannot normally find these without the now-unsupported Brent-Suyama extension. So my idea is to add an option to increase the power of each prime by 1 (or more) up to a percentage of the given value. For example, B1=1000000+50000 means that we do the first stage as normal, and then multiply E (or m) again by each prime up to 50000.
I just don't know how successful this method may be as there is no data on how many known factors there are where
k has a repeated factor greater than the square root of the suggested B1 value. James could probably run some database queries on his website and find out for us.