[QUOTE=alpertron;460355][URL="https://www.mersenne.ca/exponent/174533"]M174533[/URL] is the 314th known Mersenne number probably fully factored.[/QUOTE]

[CODE] time ../../coding/gwnum/lucasPRP M174533-cofactor 1 2 174533 -1
Lucas testing on x^2 - 4*x + 1 ...
Is Lucas PRP!

real 0m17.580s
user 0m32.212s
sys 0m19.140s
[/CODE]

[URL="http://www.mersenneforum.org/showpost.php?p=435694&postcount=14"]Reference program[/URL]

[QUOTE=ewmayer;460386]Not to seem greedy-for-credit, but a link to the aforementioned mailing-list posting might be in order for the References section. I will fwd link to this post to Chris C. after it goes up.[/QUOTE]

I think [URL="http://primes.utm.edu/top20/page.php?id=49"]Chris Caldwell's page[/URL] gives credit to the person who ran the prover program (such as Primo), i.e. the person who actually proved the cofactor was prime rather than the person who discovered that it was a probable prime.

Consistent with this, he only posts exponents with proven-prime cofactors, so his list tops out at M63703 even though the record probable-prime cofactor is for M5240707.

Even today proving primality for cofactors of an exponent of the size of M32531 takes a few weeks, so I assume it was not feasible back in 1997.

[QUOTE=GP2;460410]Even today proving primality for cofactors of an exponent of the size of M32531 takes a few weeks, so I assume it was not feasible back in 1997.[/QUOTE]
It doesn't take a few weeks. In 2017, one can have it done in a day or two and at a cost of under twenty bucks. All you need is a 32-core AWS spot instance (there are zones where it is under $10/day in spot) and need to know how to run X over ssh (which is a googlable task). Even a 15,600-digit number (~8 times more core time) is doable this way without too much trouble.

In 1997, of course, this was not feasible or cheap.

[QUOTE=GP2;460410]I think [URL="http://primes.utm.edu/top20/page.php?id=49"]Chris Caldwell's page[/URL] gives credit to the person who ran the prover program (such as Primo), i.e. the person who actually proved the cofactor was prime rather than the person who discovered that it was a probable prime.
.[/QUOTE]

The prover gets credit with the option of sharing it with the PRP'er/factorer :smile:

Yes, that's correct. For example I found that the cofactor of M19121 is PRP but I'm listed at [url]https://primes.utm.edu/primes/page.php?id=119721[/url] even though I have not ran the Primo primality program.

[QUOTE=GP2;460410]Even today proving primality for cofactors of an exponent of the size of M32531 takes a few weeks, so I assume it was not feasible back in 1997.[/QUOTE]

Back then Francois Morain and I set the record for general-prime proving with the ECPP proof of the M7331 cofactor, and that required several months-long attempts on an Alpha workstation, separated by a crucial bugfix by Morain to his ECPP code. So yeah, M32531 was way out of reach of anything below supercomputer hardware, and IIRC Morain's code was not ||-capable, anyway.

[QUOTE=paulunderwood;460434]The prover gets credit with the option of sharing it with the PRP'er/factorer :smile:[/QUOTE]

[QUOTE=alpertron;460450]Yes, that's correct. For example I found that the cofactor of M19121 is PRP but I'm listed at [url]https://primes.utm.edu/primes/page.php?id=119721[/url] even though I have not ran the Primo primality program.[/QUOTE]

Ah OK. Personally I choose to disclaim any credit.

[QUOTE=alpertron;379831]Using the formulas above I computed the expected numbers of PRP to be found in different ranges. Notice that most Mersenne numbers are not factored up to these bounds because the factorization stopped when the first prime factor was found.

[CODE]
Range Known PRPs 25 digits 30 digits 35 digits
------------------------------------------------------------
1K-2K 45 14 17 20
2K-5K 26 17 20 24
5K-10K 16 12 14 16
10K-20K 10 11 13 15
20K-50K 8 13 16 18
50K-100K 7 9 11 13
100K-200K 5 9 10 12
200K-500K 5 11 13 15
500K-1M 3 8 9 11
1M-2M 3 7 9 10
2M-3M 0 4 5 6
3M-4M 0 3 3 4
4M-5M 0 2 3 3
5M-10M 0 7 8 9
[/CODE][/QUOTE]

The above was posted in August 2014.

It represents a prediction, made at the time, about the second-largest factor of fully-factored (or probably-fully-factored) Mersenne exponents. Note that the actual largest factor, of course, is an enormous cofactor which is omitted from our factor data tables.

The above table tries to predict the number of FF-or-PFF exponents in a given range whose second-largest factor has less than 25 digits (or is that supposed to be less than or equal to 25 digits?), less than 30 digits, less than 35 digits.

Since 2014, more PRPs have been discovered, but also more ECM has been done:

All exponents up to 2389 have been ECM tested to t=50
All exponents up to 2591 have been ECM tested to t=45
All exponents up to 8017 have been ECM tested to t=40
All exponents up to 14341 have been ECM tested to t=35
All exponents up to 103067 have been ECM tested to t=30

Also, PRP testing has been done up to 6M so far. That means that no new PRP cofactors (i.e., no new FF-or-PFF exponents) will be discovered under 6M unless as a result of the discovery of new factors.

However, since ECM has been done up to t=30 for exponents up to 100K and a bit beyond, we would expect very few if any new factors smaller than 25 digits to be discovered in this range, and in any case, any individual factor discovery has very low probability of resulting in a new PRP discovery.

In other words, we already have definitive data for the top part of the "25 digits" column in the table above, and can compare it to the predictions.

I took a preliminary look, and the actual numbers seem to be running consistently below the predicted numbers.

Before posting the actual numbers, I'd like to clarify whether "25 digits" in the prediction meant "less than 25" or "less than or equal to 25".

[QUOTE=GP2;460699]Before posting the actual numbers, I'd like to clarify whether "25 digits" in the prediction meant "less than 25" or "less than or equal to 25".[/QUOTE]

I believe it is 25 digit or less (i.e. < 10^25).

We can create the table below, which reflects known PRPs rather than predicted numbers. For an explanation of the meaning of the numbers, see my post above.

The numbers towards the bottom and towards the rightmost (35 digits) column will likely increase over time, since PRP testing and ECM testing for new factors is incomplete.

However, the number towards the top and towards the leftmost (25 digits) column are unlikely to increase. For example, all exponents in the 1K to 2K range have been ECM tested to t=50, so we aren't going to find any new factors of 25 digits or less. So the current values are the final values.

The numbers do seem to be below the predicted values.

[CODE]
Range Known PRPs 25 digits 30 digits 35 digits
------------------------------------------------------------
1K–2K 52 11 12 14
2K–5K 26 10 14 21
5K–10K 17 11 12 14
10K–20K 13 6 8 9
20K–50K 12 6 9 10
50K–100K 9 6 8 9
100K–200K 8 6 7 8
200K–500K 8 8 8 8
500K–1M 6 6 6 6
1M–2M 4 4 4 4
2M–3M 1 1 1 1
3M–4M 1 1 1 1
4M-5M 2 2 2 2
5M–10M 1 1 1 1
[/CODE]

[B]1K–2K[/B]
25 digits or less: [M]1049[/M] [M]1063[/M] [M]1103[/M] [M]1223[/M] [M]1289[/M] [M]1303[/M] [M]1327[/M] [M]1459[/M] [M]1487[/M] [M]1531[/M] [M]1637[/M]
26-30 digits: [M]1307[/M]
31-35 digits: [M]1097[/M] [M]1997[/M]
more: [M]1009[/M] [M]1013[/M] [M]1019[/M] [M]1021[/M] [M]1031[/M] [M]1033[/M] [M]1039[/M] [M]1051[/M] [M]1061[/M] [M]1069[/M] [M]1087[/M] [M]1091[/M] [M]1093[/M] [M]1109[/M] [M]1117[/M] [M]1123[/M] [M]1129[/M] [M]1151[/M] [M]1153[/M] [M]1163[/M] [M]1171[/M] [M]1181[/M] [M]1187[/M] [M]1193[/M] [M]1201[/M] [M]1301[/M] [M]1321[/M] [M]1361[/M] [M]1373[/M] [M]1409[/M] [M]1427[/M] [M]1543[/M] [M]1553[/M] [M]1559[/M] [M]1657[/M] [M]1693[/M] [M]1783[/M] [M]1907[/M]

[B]2K–5K[/B]
25 digits or less: [M]2699[/M] [M]2837[/M] [M]2927[/M] [M]3041[/M] [M]3359[/M] [M]3547[/M] [M]3833[/M] [M]4127[/M] [M]4243[/M] [M]4751[/M]
26-30 digits: [M]2251[/M] [M]2447[/M] [M]2909[/M] [M]3079[/M]
31-35 digits: [M]2069[/M] [M]2311[/M] [M]2383[/M] [M]2677[/M] [M]3259[/M] [M]4729[/M] [M]4871[/M]
more: [M]2087[/M] [M]2243[/M] [M]2381[/M] [M]2549[/M] [M]4219[/M]

[B]5K–10K[/B]
25 digits or less: [M]5087[/M] [M]5227[/M] [M]5689[/M] [M]6883[/M] [M]7039[/M] [M]7331[/M] [M]7417[/M] [M]7673[/M] [M]8849[/M] [M]9697[/M] [M]9733[/M]
26-30 digits: [M]6043[/M]
31-35 digits: [M]7757[/M] [M]9901[/M]
more: [M]5233[/M] [M]6199[/M] [M]6337[/M]

[B]10K–20K[/B]
25 digits or less: [M]10007[/M] [M]11813[/M] [M]12451[/M] [M]14561[/M] [M]14621[/M] [M]17029[/M]
26-30 digits: [M]17683[/M] [M]19121[/M]
31-35 digits: [M]10211[/M]
more: [M]10169[/M] [M]10433[/M] [M]11117[/M] [M]12569[/M]

[B]20K–50K[/B]
25 digits or less: [M]20887[/M] [M]28759[/M] [M]28771[/M] [M]29473[/M] [M]32531[/M] [M]41263[/M]
26-30 digits: [M]25243[/M] [M]35339[/M] [M]41521[/M]
31-35 digits: [M]41681[/M]
more: [M]25933[/M] [M]26903[/M]

[B]50K–100K[/B]
25 digits or less: [M]57131[/M] [M]58199[/M] [M]63703[/M] [M]82939[/M] [M]86371[/M] [M]87691[/M]
26-30 digits: [M]84211[/M] [M]86137[/M]
31-35 digits: [M]53381[/M]

[B]100K–200K[/B]
25 digits or less: [M]106391[/M] [M]130439[/M] [M]136883[/M] [M]157457[/M] [M]173867[/M] [M]175631[/M]
26-30 digits or less: [M]174533[/M]
31-35 digits: [M]151013[/M]

[B]200K–500K[/B]
25 digits or less: [M]221509[/M] [M]270059[/M] [M]271211[/M] [M]271549[/M] [M]406583[/M] [M]432457[/M] [M]440399[/M] [M]488441[/M]

[B]500K–1M[/B]
25 digits or less: [M]576551[/M] [M]675977[/M] [M]684127[/M] [M]696343[/M] [M]750151[/M] [M]822971[/M]

[B]1M–2M[/B]
25 digits or less: [M]1010623[/M] [M]1168183[/M] [M]1304983[/M] [M]1790743[/M]

[B]2M–3M[/B]
25 digits or less: [M]2327417[/M]

[B]3M–4M[/B]
25 digits or less: [M]3464473[/M]

[B]4M–5M[/B]
25 digits or less: [M]4187251[/M] [M]4834891[/M]

[B]5M–10M[/B]
25 digits or less: [M]5240707[/M]

The 315th in the list is: [URL="http://www.mersenne.ca/exponent/611999"]M611999 = 18464214225958267477777390354183 * PRP184199[/URL]