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Explanation of min/mid/max P-1 bounds selection, by Mihai Preda (2020-Aug-16):

"min": is "most efficient" bounds. These are bounds such that the amount of work saved minus the amount of work invested is maximized. This maximum is reached at the point where the marginal benefit becomes equal to the marginal work (i.e. the point where a small increment of work brings an equal increment of benefit). Past this point, the benefit increment is smaller than the work increment, and the efficiency decreases.

But thinking of efficiency this way completely discards any "residual" value in a factor found (other than deciding the PRP early). But it seems we're in agreement that a factor has a value beyond the PRP that is greater than zero. The above "most efficient" bounds discard this value, acting as if the factor has zero value beyond the PRP.

One important problem with P-1 is that it's mostly "one shot", i.e. it can only be done once per exponent (mostly). So if we run the first P-1 with "most efficient" bounds, we may leave on the table some easy chance of finding a factor and lose that opportunity.

In addition to the "most efficient" bounds, I propose another notion of bounds, let's call them "max bounds". These are the bounds at which running the (P-1, PRP) combo costs just as much as running the PRP alone. The P-1 at "max" bounds does not reduce the expected time (nor does it increase it), but instead it produces factors with some probability that is higher than at "most efficient" bounds.

In terms of benefit and cost, "max bounds" are at the point where benefit==cost (while "most efficient" bounds are at the point where delta-benefit == delta-cost).

There is no point in ever running P-1 with bounds lower than the "most efficient" ones; in this sense, the "most efficient" bounds could also be called "min bounds" (the smallest bounds that make sense). It is possible to run P-1 with bounds larger than the "max bounds", but then the agent is actively investing in factor-finding at the detriment of PRP efficiency. It is also possible to run with intermediate bounds in-between "min bounds" and "max bounds", if the user has a particular balance between PRP efficiency and factor-preference.

"intermediate bounds" does not mean that any random pairing of (B1,B2) is good -- no. For any given amount of work (invested in P-1) there is a particular pair (B1,B2) that is optimal for that work.

The search proceeds like this: start from a very small (B1,B2) initial point and only move up from there. At every step, compute the delta-benefit and the delta-work for a step in the B1 direction, and for a step in the B2 direction, and proceed with the axis that has the larger ratio delta-benefit-to-delta-work. When this ratio falls below 1, we reached the "most efficient" bounds. When the total benefit falls below total work, we reached "max bounds". Any intermediate point between the two can be used as "intermediate bounds" (i.e. has an efficient balance between B1 and B2).

• min: bounds such that the amount of work saved minus the amount of work invested is maximized (delta-benefit == delta-cost)
• max: bounds at which running the (P-1, PRP) combo costs just as much as running the PRP alone (benefit==cost)
• mid: optimal midpoint between "min" and "max"