With a bit more parsing of data files, here is a list of all exponents which are listed in the mersenne.org ECM progress tables, but which Eddington has down as fully factored. The smallest, M12451 has had a considerable amount of ECM work, some of which no doubt yielded the 80 bit factor, but some of which may have been a waste of effort since that factor was found.

The 1146digit factor of M4219 has been proved by factordb.

I'm running the 1431digit cofactor of M4871 through Primo.

It's PRP to all bases=<97.

I will probably post the certificate in about six hours.

[QUOTE=lorgix;238357]The 1146digit factor of M4219 has been proved by factordb.

I'm running the 1431digit cofactor of M4871 through Primo.

It's PRP to all bases=<97.

I will probably post the certificate in about six hours.[/QUOTE]

The 1431digit factor of M4871 is definitely prime.

You missed 32351 btw (composite exponent).

The 12395digit cofactor of M41263 is PRP using all prime bases =<37.

I trust you'll submit these to Eddington.

The 63-bit factor of M14561 was probably discovered by P-1 (B1=2551, B2=461639 would do it). If this is the case, then every one of the at least 280 curves at B1=50,000 and 251 curves at B1=250,000 were a waste of time.

[QUOTE=lorgix;238373]You missed 32351 btw (composite exponent).[/QUOTE]

I was only considering prime exponents. I had assumed that the ECM Progress on the PrimeNet server, only tracked these. Is that not the case?

Appart from mprime, the only tool I have installed on my system is pari/gp, which isn't that good for this sort of thing. Do you think you will do all those feasible? Or should I install something else and help?

[QUOTE=Mr. P-1;238375]I trust you'll submit these to Eddington.

The 63-bit factor of M14561 was probably discovered by P-1 (B1=2551, B2=461639 would do it). If this is the case, then every one of the at least 280 curves at B1=50,000 and 251 curves at B1=250,000 were a waste of time.[/QUOTE]

I'm assuming this Eddington person would only be interested in the proof certificate.

Yeah, I'm pretty sure that must have been found by P-1. Damn, all the time wasted...

[QUOTE=Mr. P-1;238376]I was only considering prime exponents. I had assumed that the ECM Progress on the PrimeNet server, only tracked these. Is that not the case?[/QUOTE]

Nvm, I screwed up, typo.

[QUOTE=Mr. P-1;238378]Appart from mprime, the only tool I have installed on my system is pari/gp, which isn't that good for this sort of thing. Do you think you will do all those feasible? Or should I install something else and help?[/QUOTE]

Unfortunately, my computer is old and slow. If you're thinking about absolute proofs then I'd rather leave that to someone with more computing power. But PRPs are fast, I'm running the whole list now.

[CODE](2^6199-1)/179635392927728929353086482614805697378612611809363271
(2^6337-1)/4501318288109788011334762909785912975982707465271020857512756782053472363053668157533504357730278762590457
(2^7417-1)/7507005070637706149786475668997639973991
(2^9901-1)/126963693135213650972501160277833204460511647031933548221705424113461245441590718863886455178721617007207585872256907920259294592750741967
(2^10007-1)/20453755399107134938775648992529401
(2^10169-1)/10402314702094700470118039921523041260063
(2^10211-1)/306772303457009724362047724636324707614338377
(2^11813-1)/14740786026020930383
(2^12451-1)/87065160674565854360715728029229940740377
(2^14561-1)/8074991336582835391
(2^14621-1)/19171116593282984132708214394856581399
(2^17029-1)/418879343
(2^17683-1)/14570261281140293911854048050469358706809858630769
(2^20887-1)/118421949251433453827857391594130772157699369945439163022281
(2^26903-1)/1113285395642134415541632833178044793
(2^28759-1)/226160777
(2^28771-1)/104726441
(2^29473-1)/139640239316440423720389373549494109943
(2^32531-1)/1641222175081417
(2^41263-1)/1379707143199991617049286121
(2^41521-1)/41602235382028197528613357724450752065089
(2^57131-1)/61481396117165983261035042726614288722959856631
(2^58199-1)/237604901713907577052391
(2^63703-1)/42808417
(2^82939-1)/883323903012540278033571819073
(2^86371-1)/41681512921035887
(2^87691-1)/806957040167570408395443233
(2^106391-1)/286105171290931103
(2^130439-1)/260879
(2^136883-1)/536581361
(2^173867-1)/52536637502689
(2^221509-1)/292391881
(2^271211-1)/613961495159
(2^271549-1)/238749682487
(2^406583-1)/813167[/CODE]I've finished all but the last two. All 3-PRP so far. -edit: All 3-PRP

[QUOTE=lorgix;238385]I'm assuming this Eddington person would only be interested in the proof certificate.[/QUOTE]

A cert would allow him to promote a 'd' to a 'D'. (Alternatively a witness would convert it to a C.) It's possible, given how quickly your "old and slow" computer was able to generate a cert for cofactor:M4871, that he may already have one, and that his list is just out of date.

However it can't hurt to send it him, and your PRP confirmations, which strengthen confidence in the 'd' designation.

[QUOTE]Yeah, I'm pretty sure that must have been found by P-1. Damn, all the time wasted...[/QUOTE]

A counterargument is that the wasted curves are only a tiny fraction of a percent of the total ECM work done on all exponents, so don't really matter too much.

[QUOTE]Unfortunately, my computer is old and slow. If you're thinking about absolute proofs...[/QUOTE]

Eventually, yes, but PRPs will do for now. Do you intend to do any more that 3-PRP on them?

I suppose I should parse his lists to find the first few 'status unknown' cofactors, to see if any are small enough to be PRP-tested.

[QUOTE=lorgix;238225]Yes! I've been thinking about this for a while. We don't know what work has been done on numbers with known factors, except ECM.[/QUOTE]

Huh? The only other work that there is to do is factor it completely
with SNFS. We certainly do know what work has been done.

[QUOTE=R.D. Silverman;238469]Huh? The only other work that there is to do is factor it completely
with SNFS. We certainly do know what work has been done.[/QUOTE]

We are talking about all partially-factored prime-exponent Mersennes, not just the small ones. M8000053, for example, has a 27 bit factor, undoubtedly discovered long ago by TF. It's way too large for SNFS now or in the foreseeable future, but additional TF, P-1, or ECM might yeild more factors. Unfortunately, the database doesn't tell us how much TF beyond 27 bits or P-1, if any, has been done on this number. Nor does it tell us whether any primality or PRP tests have been done on the cofactor

You, personally, might not be interested in the discovery of additional factors short of complete factorisation. But other people are.