I'm not one of the mathematicians working on OPNs, but I'm an enthusiastic supporter who has helped coordinate work. Here is my understanding of the situation.

[B]1.[/B] Using conventional factor chains of the type seen, among other places, in the Brent & Cohen 1989 paper proving 10^160, you cannot presently exceed a lower bound of 10^568 because no factors are known for 2801^83-1. This factorization blocks the bases 3, 7, and 13. The factor chains would fail on

sigma(3^2) = 13
sigma(13) = 2 * 7
sigma(7^4) = 2801
sigma(2801^82) = C283

[B]2.[/B] Using the methods in the Brent, Cohen, and te Riehl paper 1991 paper proving 10^300, you cannot presently exceed a lower bound of approximately 10^705 because of the same roadblock.

[B]3.[/B] With the recent factorizations of 2801^79-1 and 3^607-1, plus methods that Pascal has developed that involve splitting some of the factor chains and also methods said to be inspired by Kevin Hare's work, Pascal was able to extend the lower bound to 10^1350. This limitation was a combination of the same roadblock of 2801^83-1, plus the problem that these other methods to get around the roadblock themselves floundered on 1013122723^1701 and 14713^43-1.

[B]4.[/B] With these latter two factorizations recently provided, these new methods can extend the lower bound to 10^1500.

I very sure about points 1 & 2. Points 3 & 4 reflect my understanding but could be mistaken. My guess is that a paper is in preparation for the new methods and the 10^1500 bound.

William

The remaining composites in t400.txt has been recently factored by chris2be8. Now there is [URL="http://www.lri.fr/~ochem/opn/t450.txt"]t450.txt[/URL] in which I just factored the smallest composite by GNFS:

C115 of 125731^31-1 = P57 * P58

P57 = 389505528048407043095134726941955227925051769059452436879
P58 = 7547915264801316393904683610667024458282507085484611419511

Right now I have no interest to factor any other numbers there. I prefer smaller ones. :)

How much work has been done on these and do they need any more ecm?

[QUOTE=VJS;236867]How much work has been done on these and do they need any more ecm?[/QUOTE]
I don't know, but my guess is that at least enough ECM has been done on "small" composites in t450.txt.

From Pascal's t1200.txt
[CODE]
C89 of σ(81675973663481284478214643775936930875300293735411413414107^2) =
3031536431033387074465933582851169 * P55

C95 of σ(278906118386577494263687159230870546126269722059829^2) =
13741078475378737363472259301 * 1683282458557284751919232783103 * P37

C99 of σ(98296826709875298709212297704079195775020254431544659^2) =
92136581805308631739726381469788236559 * P61
[/CODE]

From t450
[URL="http://factordb.com/index.php?id=1100000000236410743"]σ(11232875364041,12)[/URL] is factored.

Has anyone done any work on sigma(1630954578528106158097259780845944093769939317863135626697854967^2)?

It's listed in the website. If not;

I found some factors; 15864152492392273, and 3636432313.

A 101-digit composite cofactor remains.

.. Nevermind. It's already been worked on. Have any of you had any luck with larger factors so far?

Catching up on factorizations from the last couple of weeks. Except for the last one, these were

ECM to t50 by yoyo@home
SNFS sieving by RSALS
Post Processing as indicated
[code]
post processing by Zetaflux:
613^89-1
P116: 90667269484838832757904281887466089472567221237646017572049681405592015328850194320017434972062469820185570046606581
P128: 12215949879648286021693109089629389710742478750510662418417081206039220412111917011102087202228084714480610825304881952955788519

2801^67-1
P57: 655993356101321289088229961313443595606700508082875894971
P162: 306469954445296500616850599475763803506896030689678770223472947270996696904129744972155087895444999112548214892257115703816276222718827611633121731309425207186601

3221^67-1
P95: 33920493697403307279375762222099290279711974766922071545003867058360734630028424249502466242973
P124: 4828468535701152923702806300619090909733024335718523868987370421727711216365556212430560368948427753697308870714438319183437

911^83-1
P84: 643635244293261950682762930870065743710246568283613677359047774464239289190190897699
P154: 3837495624541508229287041240196986032166260186742110909038008410270880720373034725926057298350815320899214363519858626168799875814705505824397841084916647


post processing by Jeff Gilchrist:
37^149-1
P60: 486263904362387498964014190332630610262011231929421459705391
P69: 806347922351720044890009980941524851277983164589422090858327800568373
P84: 198667831419967384000989448212588102481539969521653925150385122203116238580063654113

911^83-1
P84: 643635244293261950682762930870065743710246568283613677359047774464239289190190897699
P154: 3837495624541508229287041240196986032166260186742110909038008410270880720373034725926057298350815320899214363519858626168799875814705505824397841084916647


post processing by Lionel Deboux
3169^61-1
P68: 28893860167913936851599527884605103496125024553690572895986509882569
P90: 276449356609577211434521949264498144252686009994467069924057705591895987814654424091342917

factored by ECM
131^101-1
P48: 837189189816896152648633557763431532827073372991
P126: 764072816579352468510735498937980282074762698995407550164833463141981674343449234906684203131536265257589143782024464535126427
[/code]

We are having a little festival of factorization finishing. If you want to join in the fun, there are still some 29 bit and 30 bit jobs that have finished sieving and need post processing.

[url]http://boinc.unsads.com/rsals/crunching.php[/url]

Minimum recommended resources for the 29 bit jobs 3GB of memory, dual core 64 bit system - should take 3-7 days. For the 30 bit jobs we recommend at least 64 bit Quad core system with 4 GB of DDR2 memory. That should finish in 6-10 days; DDR3 memory will finish faster.

These recent factorizations had

ECM to t50 by yoyo@home
SNFS sieving by RSALS
Post Processing by Greg Childers or Lionel Debroux
[code]

post processing by Greg Childers

727^83-1
P82: 5261767928618594916854863709232222687414047268280372989389343259045305146868111751
P143: 19632082355347097946403120580787729623235051000850800772591966087586298857674085539083758003810988893511200560683435639361988214593175733503769

13^211-1
P82: 3701016990128106745299381925855295433248992297894757817290184873593934454234719613
P112: 1716418227049050743365970223940201140778353002185114551235002126850457715843754903156810930526481431647604648779

157^103-1
P48: 290722930276414444835151244523358279077339815121
P66: 506685085131259871059413400272554306172310077701160377625814033369
P105: 317385469179766522493756700162073193004781917804834084712557084478187842634966172799087407228310007727683


post processing by Lionel Debroux

3221^61-1
P79: 7656019178391762253712321316851045559761199567183800509737255645152847758546617
P116: 28090296829519855867870346349976111459125526576891713191180614573312215715840790073243368227339037507753925758257559
[/code]

The spirit of the Festival of Finishing Factorizations is contagious! Yoyo@home finished the factorization of two numbers with ECM last night.

[code]
311^97-1
P49: 7354949365611484882164190224258088505192195304793
P185: 20189276821262046822049318588855753549555500123720543168762086520844665106736158773481862141438676656368315830986393113674583845308112996838033453224388038965583111491474616473624400187

443^89-1
P51: 820927178520532286926812270179954736828355673938817
P175: 7984707009200942169611436393244032451225897134744246004318724191641303461555241503733818624284270267857775329417658392476050269308554228094534523896632796146990165969804775141
[/code]

From t450
[URL="http://factordb.com/index.php?query=%28477869921%5E19-1%29%2F477869920"]σ(477869921^18)[/URL] is factored.