[QUOTE=Batalov;379476]GIMPS actually has all LL residues. Iirc from EWMayer, LL residues can also be used, but the interested party will have to dig up all details.[/QUOTE]
I don't believe that. Who reported full residues to GIMPS? What gimps have is the last hex digits of each residue, which can't be of any use here.

[QUOTE=ewmayer;342684]Testing the cofactor for rigorous primality is expensive, but PRP-testing (starting with the full LL residue) can be done similarly cheaply as cofactor-PRP testing is done for Fermat numbers (starting with a Pépin-test residue).

In the late 1990s I adapted the procedure used for Fermats to PRP-test a bunch of M(p) cofactors (and discovered numerous PRP cofactors up to my search limit of a then-lofty 20k digits or so). My version of the test needed a Pépin-style test residue to be generated for the target M(p), but Peter Montgomery [URL="http://ndatech.com/mersenne/archives/digest/v01_0368.txt"]showed me how to use an LL-test residue[/URL], thus eliminating the redundant LL-test/Pépin-style-mod-powering effort.[/QUOTE]
Scroll the linked discussion until "Saving storage when testing cofactors of Mp". I remembered correctly: the storage space [I]can [/I]be saved, but it is not the lowest 64bit that should have been saved but the few hundred [I]highest [/I]bits. So, as GIMPS "free-of-charge" residues go now, they probably won't help. Because LL test is a test in a weird base [TEX]\omega,\ \bar{\omega}[/TEX] (in diguise), the result is more twisted than in simple base 3. (Peter Montgomery showed how to use upper bits, ... I wonder if lower bits could be an equivalent fingerprint. Tonight I won't think about it.)

Anyway, Dario most likely has all Pépin-test residue RES64s now in his logs (up to a certain limit), so he has everything he needs for very fast new incoming factor tests within his range.

[QUOTE=Batalov;379476]All one needs to do is [U]keep all residues[/U]: 3^Mp (mod Mp) or even just their last 64 bits. [/QUOTE]

I don't see how the last 64 bits help.

Lets say F divides Mp. By Fermat, 3^(Mp/F) = 3 (mod (Mp/F)). Raising both sides to F, we get 3^Mp = 3^F (mod Mp/F). Now if we have the full residue 3^Mp (mod Mp), we can easily calculate 3^Mp (mod Mp/F). But last 64-bits? Are you saying 3^Mp = 3^F (mod Mp/F) ==> 3^Mp = 3^F (mod Mp). I just tried it with M2311 (77567729423209*4514379640917651135021865565129IPRP-652). LHS holds, but RHS doesn't.

[QUOTE=axn;379483]I don't see how the last 64 bits help.

Lets say F divides Mp. By Fermat, 3^(Mp/F) = 3 (mod (Mp/F)). Raising both sides to F, we get 3^Mp = 3^F (mod Mp/F). Now if we have the full residue 3^Mp (mod Mp), we can easily calculate 3^Mp (mod Mp/F). But last 64-bits? Are you saying 3^Mp = 3^F (mod Mp/F) ==> 3^Mp = 3^F (mod Mp). I just tried it with M2311 (77567729423209*4514379640917651135021865565129IPRP-652). LHS holds, but RHS doesn't.[/QUOTE]

Anyway, for numbers in this range, it is easier to perform a PRP test than to find a new prime factor of the Mersenne number. There is no double check of PRP, so there is still the doubt of having incorrect residues. This is why I would prefer to perform a PRP test on the new cofactor.

Did some rudimentary math on the PRP data @mersenne.ca for the odds of finding a positive PRP result given the number of factors found (three or less versus four or more).

Below M(2M) there are 3384 exponents with 4 or more factors, of these 78 have a positive PRP result.
~2.3% chance of a positive prp result given four or more factors.

For three factors or less: total of ~110801 exponents factored, subtract exponents with more than four factors leaves us with ~107417.

There are 203 positive PRP results for 3 factors or less, gives:
~0.19% chance of a positive PRP result given three or less factors.

[B]So the odds of finding a positive PRP with more than three factors is about ten times higher than for exponents with three factors or less.
[/B]
This analysis does not differentiate between one, two and three factors. The data is not readily available but I get the feeling that this would further lower the odds of a positive PRP for a single factor.

Either way it seems that running PRPs on exponents with numerous factors is more fruitful than exponents with a single, or couple, of factors.

EDIT: there seem to only be ~75 positive PRP tests for a single factor. This is only ~25% of the positive PRP tests found to date. Yet the large majority of exponents only have one factor.

[QUOTE=flagrantflowers;379686]Did some rudimentary math on the PRP data @mersenne.ca for the odds of finding a positive PRP result given the number of factors found (three or less versus four or more).
[/QUOTE]

No, you did NOT do any math. What you did was mindless numerology
on a small sample set and then jump to some clearly erroneous
conclusions. Your conclusions, if correct would violate Erdos-Kac.

Stop babbling. Stop the pseudo-math. Stop pretending that what you
are doing is math. Stop equating mindless numerology with "performing
rudimentary math".

If, in fact, you think that you did some math, please show it to us.
What you have presented is not math and does not constitute meaningful
analysis of the data.

[QUOTE=flagrantflowers;379686]Did some rudimentary math on the PRP data @mersenne.ca for the odds of finding a positive PRP result given the number of factors found (three or less versus four or more).

Below M(2M) there are 3384 exponents with 4 or more factors, of these 78 have a positive PRP result.
~2.3% chance of a positive prp result given four or more factors.

For three factors or less: total of ~110801 exponents factored, subtract exponents with more than four factors leaves us with ~107417.

There are 203 positive PRP results for 3 factors or less, gives:
~0.19% chance of a positive PRP result given three or less factors.

[B]So the odds of finding a positive PRP with more than three factors is about ten times higher than for exponents with three factors or less.
[/B]
This analysis does not differentiate between one, two and three factors. The data is not readily available but I get the feeling that this would further lower the odds of a positive PRP for a single factor.

Either way it seems that running PRPs on exponents with numerous factors is more fruitful than exponents with a single, or couple, of factors.

EDIT: there seem to only be ~75 positive PRP tests for a single factor. This is only ~25% of the positive PRP tests found to date. Yet the large majority of exponents only have one factor.[/QUOTE]

Notice that in the range you are considering, the lowest numbers have much more factoring effort done, so I think the statistics are not correct. It is like comparing apples and oranges. For example, if you consider the range 500K to 2M you will find a completely different picture.

[QUOTE=alpertron;379712]Notice that in the range you are considering, the lowest numbers have much more factoring effort done, so I think the statistics are not correct. It is like comparing apples and oranges. For example, if you consider the range 500K to 2M you will find a completely different picture.[/QUOTE]

It isn't even that! He babbles about numbers with 3 factors, vs. numbers
with 4 factors, when all he is looking at is numbers with 3 SMALL factors;
i.e. almost all of the numbers with three factors are not even included in
his data because most of these have factors whose size lie beyond his
search range. What he is doing is MEANINGLESS.

There is a formula on section 4.5.4 (Factoring into Primes) of Knuth's TAOCP that computes the probability of having the second largest prime factor less than some bound (expressed as log B / log N, B is the bound, N is the composite number).

That can be used here, but it appears that the formula is not easy to compute. For example, we could fix the bound to 10[SUP]30[/SUP] and then for all prime exponents between 1M and 2M, add all probabilities. That sum could be (approximately) the expected number of PRP factors in the mentioned range that we could found in a reasonable amount of time.

[QUOTE=alpertron;379712]Notice that in the range you are considering, the lowest numbers have much more factoring effort done, so I think the statistics are not correct. It is like comparing apples and oranges. For example, if you consider the range 500K to 2M you will find a completely different picture.[/QUOTE]


I considered ignoring <M(10K) as you're correct there are few exponents with more than one factor and the picture does drastically change. But if you ignore that range the odds of positive PRP results as a function of factors is more disparate between 1-3 and more than four.

I find it interesting that there are obviously much fewer (as a percent) of exponents with less than three factors versus more than four.

You're correct I didn't do any "math". I calculated some uncorrected percentages. The order of magnitude difference, despite my "pseudomath", seems to be too large to simply be dumb "numerology".

[QUOTE=R.D. Silverman;379714]It isn't even that! He babbles about numbers…[/QUOTE]


Unfortunately we only have PRP tests completed up to M(2M) so going beyond that range doesn't make sense as I would be including mostly candidates without a PRP result (positive or negative).

All I'm saying is the percent of exponents with a positive PRP result and having more than three factors is a very large portion of positive tests. Despite the fact that there are much more numbers with one factor.

I'm not trying to say that this means anything. I'm trying to say that this is functionally what the odds are on the tests done to date. This is obviously from this sample set and doesn't, ideally, represent the population of all <M(2M).

I don't pretend to understand the underlying factor theorems.