[QUOTE=retina;436572]Do we know that all candidates within the bit ranges are being tested?[/QUOTE]
Even if they're, it could well be that a new bit levels are started even before current one has finished. That makes it tricky to calculate the rate just from the bit completion milestones alone.

It is also possible that they're specifically only testing those factors whose P-1 is divisible by a range of primes (say 1e4 < p < 1e9). Such a thing could be efficiently implemeted by a sieve, for example.

/more-guesswork-and-speculations

[QUOTE=axn;436582]Even if they're, it could well be that a new bit levels are started even before current one has finished. That makes it tricky to calculate the rate just from the bit completion milestones alone.

It is also possible that they're specifically only testing those factors whose P-1 is divisible by a range of primes (say 1e4 < p < 1e9). Such a thing could be efficiently implemeted by a sieve, for example.

/more-guesswork-and-speculations[/QUOTE]

they could also be using the fact that you can substitute the multiply by 2 by any power of two and test higher ones with a slight alteration of method using the same basic bits and do them in parallel.

From what I remember when I was following this their algorithm only does candidates larger than M(2^x) somewhere around 2^20. This is why they post ECM results for smaller exponents weekly (guess!).

It also seems as though they have worked ahead of the factors they post daily. They typically post first factors of a bit level higher a month or so after completing the previous bit (guess & assumption?).

The rate of factors posted per day has also declined over time. I think initially it was 20k per day and is now 6k.

After 5 months of work, TJAOI found all missing 63-bit prime factors of Mersenne numbers. The last entry submitted was for [URL="http://www.mersenne.org/report_exponent/?exp_lo=470566403&full=1"]M470566403[/URL] which has 62,999999498... bits.

[QUOTE=alpertron;447614]After 5 months of work, TJAOI found all missing 63-bit prime factors of Mersenne numbers. The last entry submitted was for [URL="http://www.mersenne.org/report_exponent/?exp_lo=470566403&full=1"]M470566403[/URL] which has 62.999999498... bits.[/QUOTE]
After 7 months, TJAOI completed the 64-bit range. The last factor he submitted is the second known factor of [URL="https://www.mersenne.org/report_exponent/?exp_lo=25481647&full=1"]M25481647[/URL] which has 63.999998963... bits.

Maybe it is time for a little status update about missed factors (that were found by TJAOI or other users). (With no claim to be exhaustive.)

Anything new there? (i.e. factors for mersenne numbers that had no known factors before)

[QUOTE=LaurV;465768]Anything new there? (i.e. factors for mersenne numbers that had no known factors before)[/QUOTE]

Not "new". Many of the factors I found were from redoing TF done on a bad machine, plus a few from bit levels that were skipped. But that work was all from a year or more ago.

How systematic is TJAOI? Let's try to answer that question.

Various users in this thread have monitored TJAOI's activity, and there have been multiple posts that have proclaimed that "TJAOI has completed N bits as of date XXXX-XX-XX". A [URL="http://www.mersenneforum.org/showthread.php?t=19014&p=436550"]list of these milestones and dates[/URL] was compiled by alpertron.

We can actually check if this is true, because each factor reported to the PrimeNet database since 2008 or so has a datestamp.

The table below shows the most recent date that a factor of size N bits was added to the database. This takes into account all factors for all exponents up to the maximum exponent size of 999,999,999.

The third column is the date "proclaimed" (if any) in the various posts to this thread, with a link to the post in question.

[B]44[/B] 2014-01-10
[B]45[/B] 2014-01-13
[B]46[/B] 2014-01-31
[B]47[/B] 2014-02-21
[B]48[/B] 2014-03-21
[B]49[/B] 2014-10-06 (actually 2014-04-17 for all but one factor)
[B]50[/B] 2014-10-06 "[URL="http://www.mersenneforum.org/showthread.php?t=19014&p=373637"]2014-05-16[/URL]"
[B]51[/B] 2014-06-13
[B]52[/B] 2014-07-18 "[URL="http://www.mersenneforum.org/showthread.php?t=19014&p=378875"]2014-07-18[/URL]"
[B]53[/B] 2014-08-31 "[URL="http://www.mersenneforum.org/showthread.php?t=19014&p=381803"]2014-08-31[/URL]"
[B]54[/B] 2014-10-26 "[URL="http://www.mersenneforum.org/showthread.php?t=19014&p=386171"]2014-10-26[/URL]"
[B]55[/B] 2014-12-30 "[URL="http://www.mersenneforum.org/showthread.php?t=19014&p=391247"]2014-12-30[/URL]"
[B]56[/B] 2015-02-27
[B]57[/B] 2015-05-15 "[URL="http://www.mersenneforum.org/showthread.php?t=19014&p=402532"]2015-05-18[/URL]"
[B]58[/B] 2015-07-31 "[URL="http://www.mersenneforum.org/showthread.php?t=19014&p=407169"]2015-08-03[/URL]"
[B]59[/B] 2015-10-19 "[URL="http://www.mersenneforum.org/showthread.php?t=19014&p=412849"]2015-10-16[/URL]"
[B]60[/B] 2015-12-30 "[URL="http://www.mersenneforum.org/showthread.php?t=19014&p=420515"]2015-12-30[/URL]"
[B]61[/B] 2016-03-22 "[URL="http://www.mersenneforum.org/showthread.php?t=19014&p=429782"]2016-03-22[/URL]"
[B]62[/B] 2016-06-18 "[URL="http://www.mersenneforum.org/showthread.php?t=19014&p=436527"]2016-06-18[/URL]"
[B]63[/B] 2016-11-28 "[URL="http://www.mersenneforum.org/showthread.php?t=19014&p=447614"]2016-11-23[/URL]"
[B]64[/B] 2017-06-21 "[URL="http://www.mersenneforum.org/showthread.php?t=19014&p=461679"]2017-06-21[/URL]"

Factors of size 65 bits and higher keep being found as recently as today.

There are only a handful of exceptions where the actual date of the most recent factor(s) of a given bit size was after the "proclaimed" date. Those are shown below. All were discovered by TJAOI, except for the factor associated with the smallest exponent 787757, which was discovered by alpertron.

49 bits
51024277,562830736992943,2014-10-06

50 bits
787757,1125287845211863,2014-10-04 [B]**** discoverer: alpertron[/B]
143587487,1125854839643327,2014-10-06
266371537,1125890155959343,2014-10-06
523242581,1125888270151913,2014-10-06
66227927,1125894892289809,2014-10-06
707889751,1125789633943847,2014-10-06
930639257,1125887373118601,2014-10-06

59 bits
210271813,397255992431108929,2015-10-19
406392263,479585916221281487,2015-10-19

63 bits
486184849,6124891029543354631,2016-11-28


For 59 bits and 63 bits the discrepancy is only a few days, and only one or two straggler factors.

For 49 bits, no "proclamation" was made in this thread. There is only one outlier that arrived almost six months after the rest.

For 50 bits, there were seven factors reported almost five months after the proclamation date.

The only factor that was apparently missed by TJAOI was the factor 1125287845211863, which has a bit size of 49.9995, extremely close to the bit-size boundary. However, this discovery by alperton must have prompted TJAOI to find and fix a bug in his code, because two days later there were six new factors reported for bit size 50 and one new factor reported for bit size 49, all by TJAOI, and all extremely close to a bit-size boundary (note 2[SUP]49[/SUP]=562949953421312 and 2[SUP]50[/SUP]=1125899906842624.

After this one glitch three years ago, which appears to have been fixed, it really does hold empirically that when TJAOI finishes with a particular bit-size range, no new factors of that bit size get discovered, by him or by anyone else.

We conclude that TJAOI's methodology is indeed systematic, and trial-factoring for ranges below 65 bits will almost certainly be completely fruitless. We might expect TJAOI to complete the 65-bit range sometime in 2018.

What is the lowest exponent for which TJAOI found a factor?

[QUOTE=Mark Rose;467995]What is the lowest exponent for which TJAOI found a factor?[/QUOTE]

That's not so easy to answer, because the [URL="https://mersenne.org/report_factors/"]Factors Found[/URL] report doesn't include discoverers. For any given exponent, the factor discoverers are usually listed in the History section of the Exponent Status report.

Maybe you could look at the archives of the daily XML report, but I forget the URLs for those.