[QUOTE=swellman;336724]Do you still have interest in the C488 and C572 of this thread?[/quote]

Definitely. The C243 also.

[quote]Just might take awhile.[/QUOTE]

Much appreciated.

Just to clarify, do you mean the C243 associated with the 26th term of this series?

If so, it seems it was verified in [url=http://www.mersenneforum.org/showpost.php?p=334171&postcount=3]this post[/url]. Or do you want to factor it for completeness?

Either way it's on the list!

[QUOTE=swellman;336804]Just to clarify, do you mean the C243 associated with the 26th term of this series?

If so, it seems it was verified in [url=http://www.mersenneforum.org/showpost.php?p=334171&postcount=3]this post[/url].[/quote]

Also with his P-1 trick [url=http://mersenneforum.org/showpost.php?p=334843&postcount=41]here[/url].

[quote]Or do you want to factor it for completeness?[/quote]

For completeness yes, though I will confess to feeling slightly uneasy about the P-1 trick. I can't think of any reason why it shouldn't work, but it's not as straightforward conceptually as traditional trial division. (Actually I can think of a reason - operator error. It's a bit more complex to do than just sticking in the -t flag to gmp-ecm.)

I have similar reservations about Pollard-Strassen. Also, to my knowledge arbooker's code hasn't been subject to the same degree of peer review as has the gmp-ecm codebase, no disrespect to him of course.

[QUOTE]Either way it's on the list![/QUOTE]

Appreciated, though obviously it should be toward the bottom. [url=http://mersenneforum.org/showpost.php?p=336811&postcount=11]Three cofactors from my other list[/url] can be deprioritised for the same reason. Could I beg a further addition as a high priority, namely [url=http://factordb.com/index.php?query=24741510861371072371859887195453971978233358463487924273876239247176461296447003498710103623746010339740673425357288675718589856208972430477774351068955749733538298909424289611414118755150707490315356665410733273230021227105770181577517840539729404418635841415399934101807914846564754342241742437572074594138915764312819]this C276 cofactor[/url]? The ostensibly smallest congruent factor is a p39, so a mere t45 won't make me feel as confident as it does with smaller factors..

Don't start anything from that other thread just yet. I want to doublecheck all my calculations so far.

[QUOTE=Mr. P-1;336814]
Appreciated, though obviously it should be toward the bottom. [url=http://mersenneforum.org/showpost.php?p=336811&postcount=11]Three cofactors from my other list[/url] can be deprioritised for the same reason. Could I beg a further addition as a high priority, namely [url=http://factordb.com/index.php?query=24741510861371072371859887195453971978233358463487924273876239247176461296447003498710103623746010339740673425357288675718589856208972430477774351068955749733538298909424289611414118755150707490315356665410733273230021227105770181577517840539729404418635841415399934101807914846564754342241742437572074594138915764312819]this C276 cofactor[/url]? The ostensibly smallest congruent factor is a p39, so a mere t45 won't make me feel as confident as it does with smaller factors..

Don't start anything from that other thread just yet. I want to doublecheck all my calculations so far.[/QUOTE]

So the C276 is currently your top priority?

[QUOTE=swellman;337005]So the C276 is currently your top priority?[/QUOTE]

It was the one I most wanted to see taken up to t50. I say "was" because since I posted, I've unblocked sequence A217759 by pulling another 3 (mod 4) P39 out of a minus side algebraic factor. For the same reason as the C276, I now want to see both the [url=http://factordb.com/index.php?query=5219617398952013472248459322513064797141595503550208672004049535341256097641721558874167306231533888374664511754837051589505670453625585201723618497816402437013188831582203838813202311972735192075432804452341222518325876997834560978921892864195202526546951306285972510053911496882145499872159495292617309924703284463058107240904177024981942527908750099745126240683794523736317745]C329 cofactor[/url] and the as-yet unbroken plus-side algebraic factor, a [url=http://factordb.com/index.php?query=5219617398952013472248459322513064797141595503550208672004049535341256097641721558874167306231533888374664511754837051589505670453625585201723618497816402437013188831582203838813202311972735192075432804452341222518325876997834560978921892864195202526546951306285972510053911496882145499872159495292617309924703284463058107240904177024981942527908750099745126240683794523736317747]C379[/url] taken to t50.

I'm currently working on all three. None is even a fifth of the way into a t45. Moreover I really do want to check the calculations leading up to these numbers. I'd hate to have you expend large amounts of CPU power on my behalf, only for it all to have been wasted because I screwed up somewhere.

Edited to add: There's also the [url=http://factordb.com/index.php?query=(2*5*101*1020101*53*29*2507707213238852620996901*449*13*8693*1997*6029*61*3181837*113*181*1934689*6143090225314378441493352126119201470973493456817556328833988172277*4733*3617*41*68141*37*51473*17*821*598201519454797*157*9689*2357*757*149*293*5261*1774915226229214328737*233*73*13989077153*11689487473519919005062249197*353*4021*24891406347771253321)^2%2B1]C572[/url] currently blocking A057207, the subject of this thread, which I have mixed feelings about. If you took it to t50 and found nothing, then we'd conclude that continuing this sequence was beyond the resources we're prepared to devote to it. But what if you found a P50? would you be prepared to go on to t55 which would, I think, be the minimum necessary to be confident that it was the smallest? What if the factor you found was even larger? The P53 I found [url=http://mersenneforum.org/showpost.php?p=334810&postcount=4]here[/url] was eight digits larger than the t-level I was working at.

My inclination is to stay the course on the C488 and the C572 by taking them both up to t50. If an awkwardly sized factor is found, a new strategy can be devised. If no factors are found, then yes it's getting into deep waters beyond the practical limits of my little "pico farm".

The odds of detecting a factor between 45 and 50 digits is ~6.3%, correct? [i.e. (50-45)/50 *(1-e^(-1))]

Is there a Bayesian component to this calculation?

Edited to add: I will be glad to help with factoring the composites for the other sequences you are developing, just can't commit today.

[QUOTE=swellman;337098]My inclination is to stay the course on the C488 and the C572 by taking them both up to t50. If an awkwardly sized factor is found, a new strategy can be devised.[/quote]

A new factor from the C488 won't cause any problems, since we already have a small one.

[quote]The odds of detecting a factor between 45 and 50 digits is ~6.3%, correct? [i.e. (50-45)/50 *(1-e^(-1))][/quote]

I don't think so. The probability of finding a 50 digit factor [i]if there is one[/i] after a t50 is about 1-e[sup]-1[/sup], but this depends upon the size of the factor: The probability is greater if it is close to 10[sup]49[/sup], less if it is close to 10[sup]50[/sup]. The probability of finding a 45-50 digit factor [i]if there is one[/i] will vary from about 1-e[sup]-1[/sup] to about 1-e[sup]-(7557/1286)[/sup] assuming B1=43M. (The values in the ratio in the exponent were taken from ecm -v) The upper limit, which applies to 45-digit numbers, is about 99.7%

The probability of finding a factor in that range, therefore, will be the probability that a factor of size s exists, multiplied by the probability of finding it, integrated between 10[sup]44[/sup] and 10[sup]50[/sup] (or between 44 and 50, or between 44*log[sub]2[/sub](10) and 50*log[sub]2[/sub](10), depending upon how you define "size s"). There are complex formulas estimating both of these probabilities. Bob Silverman will likely appear to recommend his joint paper with Sam Wagstaff.

[quote]Is there a Bayesian component to this calculation?[/quote]

You had to ask, didn't you? We know that N is composite, i.e., that it has at least one factor less than sqrt(N), that there are no very small factors (below the TF limit), and that, because of the prior ecm, the probability of a small factor above that limit is low. More and more ecm will tend to shift the expected size of the smallest factor upwards, but it can't push it past sqrt(N). Small factors become less likely, large factors moreso. Exactly where balance point is depends not just upon the t level, but also the size of N. You might think that a t50 without finding a factor will always reduce the likelyhood that a 50 digit factor exists, but imagine doing one on a C100. The maximum possible size of the smallest factor is 50 digits, so in this case the likelyhood that it is exactly 50 digits must increase.

It's complicated.

[quote]Edited to add: I will be glad to help with factoring the composites for the other sequences you are developing, just can't commit today.[/QUOTE]

I understand.

[QUOTE=Mr. P-1;337070]I'd hate to have you expend large amounts of CPU power on my behalf, only for it all to have been wasted because I screwed up somewhere.[/quote]

[url]http://mersenneforum.org/showpost.php?p=337215&postcount=12[/url] :blush:

Measure twice, cut once.:smile:

Of course I'm an awful carpenter...

C488 ECM'd to t50
 

7600 curves with B1=43M on the [url=http://factordb.com/index.php?id=1100000000593234891]C488[/url] with no factors found. :bangheadonwall:

Releasing the number. Sorry.

Currently running ECM on the C243 and the C572.

Ran ECM on the C243 for 7600 curves with B1=43M. No factor found.

Releasing the number. Sorry.

eta:Still working the C572.