PRP testers get 'em while they are hot. ~35 new tests are up all under M(1M) some under M(100k).

EDIT: You can thank my hero TJAOI for those btw.

After executing about 20000 PRP tests I've just completed the range of Mersenne Numbers with exponents less than 2 million as can be seen on [url]http://www.mersenne.ca/prp.php?assigned_distribution=1[/url] .

This range will be complete only for a short time because every day there are new known prime factors of Mersenne numbers with exponents less than two million.

Great work, alpertron! You found couple of nice gems on the way. I think that even for a year of factors in this range (below 2M) the new PRP tests needed will be negligible work. Any plans to go to 3 :smile:?

[QUOTE=bloodIce;379411]Great work, alpertron! You found couple of nice gems on the way. I think that even for a year of factors in this range (below 2M) the new PRP tests needed will be negligible work. Any plans to go to 3 :smile:?[/QUOTE]

Thanks, but completing the 2M to 3M range will need a lot of time (a few CPU years). That should be done by several people. When done, I think that at least one PRP will be discovered.

Alperton, would you consider running PRP test on 2-3M for exponents that have 4 or more factors? ~650 tests.

[QUOTE=bloodIce;379411]new PRP tests needed will be negligible work. Any plans to go to 3 :smile:?[/QUOTE]

You might be surprised! User TJAOI is posting a lot of ECM factors (~1475 total to date) all under 1M, ~1100 in since June 1st. A lot of multiple factors for exponents, but I would expect >100 tests a week depending on progress, in addition to a few other users that seem to regularly post a couple a week.

[QUOTE=flagrantflowers;379447]Alperton, would you consider running PRP test on 2-3M for exponents that have 4 or more factors? ~650 tests.
[/QUOTE]
These are the Mersenne numbers that are the product of prime numbers and known PRP:

[QUOTE=http://www.mersenne.ca/prp.php]
M130,439 = 260879 x PRP-39261
M136,883 = 536581361 x PRP-41198
M157,457 = 4612545359 x 358012521626153 x PRP-47376
M173,867 = 52536637502689 x PRP-52326
M221,509 = 292391881 x PRP-66673
M271,211 = 613961495159 x PRP-81631
M271,549 = 238749682487 x PRP-81734
M406,583 = 813167 x PRP-122388
M432,457 = 1672739247834685086279697 x PRP-130159
M576,551 = 4612409 x 64758208321 x 242584327930759 x PRP-173528
M684,127 = 23765203727 x PRP-205933
M696,343 = 11141489 x 36009913139329 x PRP-209600
M1,010,623 = 12602017578957977 x PRP-304212
M1,168,183 = 54763676838381762583 x PRP-351639
M1,304,983 = 52199321 x PRP-392832
[/QUOTE]

So it appears that it is not more probable that the cofactor is PRP if the Mersenne number has many prime factors.

[QUOTE=alpertron;379457]So it appears that it is not more probable that the cofactor is PRP if the Mersenne number has many prime factors.[/QUOTE]

But what percentage of your candidates only have one factor? In other words, what is the percent of positive prp results for all single factors exponents, two factors, three, etc?

In your small set there are two, 2 factor exponents and one 3 factor. There are much less exponents with 2-3 factors than a single factor.

[QUOTE=flagrantflowers;379462]But what percentage of your candidates only have one factor? In other words, what is the percent of positive prp results for all single factors exponents, two factors, three, etc?

In your small set there are two, 2 factor exponents and one 3 factor. There are much less exponents with 2-3 factors than a single factor.[/QUOTE]

I went to [url]http://www.mersenne.ca/prp.php?show=2&min_exponent=900000&max_exponent=1000000[/url] (the first 1000 Mersenne numbers with exponents greater than 900,000) and then selected all lines, copied them and then pasted them in a text editor, deleting all extra lines at the top and botom.

The number of lines is 3427. Removing 1000 lines that contain Mxxx,xxx and 1000 lines of "no", we get 1427 lines, that is, there are 1427 factors, so there are many Mersenne numbers with at least 2 prime factors in that range.

Is this the appropriate place to ask about the PRP stuff?

As far as I can tell PRP refers to Probably Prime which, not too surprisingly, is a number that fits enough criteria of prime numbers (relative to a base, so probable prime base 2 or base 13 or whatever?) that we believe it to be a prime number, but we're not sure?

When finding factors, we stumble across some little ones but we're some times left with a gigantic number and we don't know whether it's prime or not (just like the Mersenne numbers) or what its factors are (again just like the Mersenne numbers).

So is this PRP business part of the ECM effort to factor the smaller Mersennes?

For the [U]size[/U] of these numbers, if they pas a few-bases PRP test, then the chances are very high that they are prime. I mean "very high" like in 99 followed by a dot, and a very long string of nines, so long it won't fit your screen line, and then the percent sign at the end. This is because the probability of a PRP to be PSP (pseudoprime, i.e. composite) decreases with the size of the number, larger PRPs have lower probability to be composite.

But we can't be sure till somebody proves them to be prime, most likely never, as proving them prime is not very important. More important could me if someone proves a theorem like "a composite factor of a mersenne number with prime exponent can not be 3-PRP", for example. (it won't work for 2, for example 2047 is 2-PRP, and yet composite). Or, ""a composite proper factor of a mersenne number with prime exponent can not be 2-PRP". But these fall more in the "open questions" (they even can't be called "conjectures") and we have no idea if they are true. There is no reason why they would be true, in fact. So, some PRP factor of a mersenne number with prime exponent can still be composite. We are talking here about numbers with thousand digits at least, ans many people fail to imagine the [U]size[/U] of those numbers. A number with 100 digits will be enough to write down the number of all atoms which exist in the known universe.

People are doing that (finding PRP's of this type) for fun.

Related to the ECM part, one can try to ECM such PRPs but her chances to get a factor are close to zero. Like in a zero, followed by a dot, followed by a hundred zeroes, followed by a 1, then the percent sign.

On the other hand, some ECM factors are found daily, for OTHER mersenne numbers in that range, and that leaves new [U]cofactors[/U] available for PRP testing. For example, if I test now the cofactor of Mx, where the exponent x is a prime in the range, and I find it to be composite (NON-PRP), then the job is not "done" yet. Someone can find a factor of it tomorrow, so I would have to come back to the range and test if the new cofactor is still composite, or not (a new PRP test). That is what the people here are talking about.

[QUOTE=LaurV;379475]For example, if I test now the cofactor of Mx, where the exponent x is a prime in the range, and I find it to be composite (NON-PRP), then the job is not "done" yet. Someone can find a factor of it tomorrow, so I would have to come back to the range and test [COLOR=Red]if the new cofactor is still composite[/COLOR], or not ([COLOR=Red]a new PRP test[/COLOR]).[/QUOTE]
All one needs to do is [U]keep all residues[/U]: 3^Mp (mod Mp) or even just their last 64 bits.

When a new factor F is reported, one can check if this new factor leads to a PRP almost instantly: calculate 3^prod(F[SUB]i[/SUB]) (mod Mp) and if it matches, do the PRP test (and probably best in a different base, because for base 3 it is already almost surely in the hat).

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.