Some time ago I raised the question on this list, whether the client's choice of the E parameter was optimal in P-1 calculations. I gave a somewhat handwavy argument in support of the claim that IF it is worthwhile choosing E=4 over E=2, i.e., if the benefits in additional factors found outweigh the cost in extra processing time[1], THEN it should also be worthwhile choosing E=6, and maybe even E=8 or E=12. I also argued on empirical grounds that, for D=420 at least, E=4 and E=12 would need to yield roughly 2% and 10% respectively more stage 2 factors than E=2 for this to be worthwhile.[2]
>From a theoretical point of view, Peter Montgomery's thesis, as suggested by Alex Kruppa, is clearly relevant. (I don't have the URL to hand, but someone is sure to post it.) Unfortunately it's somewhat beyond my mathematical ability to grasp. Therefore, I have concentrated on attempting to collect empirical evidence, as follows. I have done a great many P-1s of doublecheck assignments with E=4. Out of the 35 stage 2 factors found[3], 1 was 'extended', i.e, would not have been found using E=2. In the hope of more quickly collecting data, I have also redone, to 'first time test' limits, every entry in pminus1.txt which had previously done to B1=B2=1000, 2000, and 3000. For these exponents, all in the 1M-3M ranges, the client was able to choose a plan with E=12. Unfortunately, I found far fewer factors in either stage 1 or stage 2 than I would expect, which suggests to me that exponents in this range have had additional factoring work (possibly ECM) not recorded in the file. Of particular concern is the possibility that in addition to reducing the number of factors available for me to find, it may have upset the balance between 'normal' and 'extended' P-1 factors - the very ratio I am trying to measure. Consequently I am inclined to exclude these results, though I report them for completeness: Of the 10 stage 2 factors found, 2 were extended. They are:- P-1 found a factor in stage #2, B1=20000, B2=395000. UID: daran/1, M1231753 has a factor: 591108149622595096537 591108149622595096537-1 = 2^3*3*11*743*2689*909829*1231753 P-1 found a factor in stage #2, B1=30000, B2=547500. UID: daran/1, M2008553 has a factor: 9050052090266148529 9050052090266148529-1 = 2^4*3^2*7*71*79*796933*2008553 Finally, with George's permission, I have done a small number of P-1s of doublechecking assignments with a client modified to use D=420, E=12 - a plan not available with the standard clients. So far, I have found only one stage 2 factor, which was not extended. I will continue to search for more. Of particular interest with E=12 extended factors, is whether they would have been found with a lower value of E. E=12 will find all factors that E=4 and E=6 would have found, and some not found by any lower E. My handwavy argument predicted that E=6 should yield on average twice as many extended factors than E=4. I'm hoping that someone (Alex Kruppa?) might have a tool to analyse extended factors to determine their minimal E. If not, I will write one. In conclusion, the evidence I have been able to gather, though statistically insignificant, does not tend to exclude the hypothesis that a higher E would be worthwhile. [1]There is also a memory cost, but this is low in comparison with the costs associated with the D parameter. For example, for an exponent in the 7779000-9071000 range, in which I am working, D=420, E=4 consumes 446MB, and because of the client's conservative programming, 465MB must be 'allowed' before it will choose this plan. The next level down is D=210, E=4 which requires 299MB. Using the modified client with E=12 adds an extra 37MB to these requirements, which is memory available and going spare if the amount allowed is between about 350MB and 465MB. Another way to look at this is to say that there is no memory cost associated with increasing E for a given value of D. The memory is either available, or it is not. [2]Assuming that the current algorithm for determining optimal B1 and B2 values are accurate, and that this routine would be modified to make it aware of the costs and benefits of differing values of E. [3]This total includes both prime components of a composite factor found in a single P-1 run, since neither was extended. Regards Daran _________________________________________________________________________ Unsubscribe & list info -- http://www.ndatech.com/mersenne/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
