Here are a few results from an overnight of P-1:
[code]
sigma(702364765541756394992455108362536965996332497312551403^2)
588616471
622487694030001
770087442438799909686170818304671183
575675065910285096484096944488088577124965793

sigma(5559248220240203628717111169007640020521484534972045780770594364801987018198160426637^2)
948821001844508212039513
4338334404444849404091319
C116

sigma(91209336264878177762269108118591104597403747877718061^4)
5126204168186504784011
384438417906769411062171011
C128

sigma(638354256865636401729437150252945762392318050976276153295041317691197560244957704129069907641^2)
3643665950199368949975331
1342574547210404822677563091711
2288662683954500058160910732464954968430457457581373105135908374315564598214465479002575625151799581770804502829760397246413

sigma(2600448731324969036687295712106520101349308046307782769^4)
1467634664049003200645178477206831854961
C144

sigma(1121918318866973884164632620888225657^6)
14389849971104145218873209912554448433
105495122731457002141936386265532454423014192911232780681391117381728760032186079236825950556864490135896269744578897984840613331016107092295713446680607821

sigma(2780923^42)
355239242354818091159702308038816017
16811249996063848196335441374624029913706672411633705295995107484754713966609111482828026748327531891560817190461539432619244038589385443238129306217367423447090951

sigma(2130245071889163261645457707472426617398673017^6)
2673708075213866786517184147474737584033749
C171
[/code]

I'm now doing the two from t380.txt I've had reserved for an embarrassingly long time.

sigma(280257881^18):
r1=1038175342096359910438215859723267663686397 (pp43)
r2=3436242179333057172025576819569574559269943645252993272846012423014638342367359895330720957866758639 (pp100)

Not an ECM miss, ECM to 45 digits would have taken much longer than SNFS.

Chris K

Reserving t400.txt
 

After factoring the last number from t380.txt I intend to factor the numbers in t400.txt. They should only take a few days each with SNFS.

This looks a good project for autumn.

Chris K

From t400.txt:
sigma(2699538733^18)
r1=1807994650694311985393069324496482467528871470498655813801278879695731 (pp70)
r2=2097416632624700864382059774029576360330939924894360263560724710498728564841036853111929 (pp88)

Chris K

Finally the factorization of the roadblock 251^103-1 = P69 * P177. The post processing was delayed by scheduled and unscheduled system outages and summer travel schedules that seemed to always fall in such a way as to maximize the delay.

ECM to t50 by yoyo@home
SNFS sieving by RSALS
Post Processing by Pace Nielsen

[code]
The roadblock was

sigma(127^30) = 1310825268269643509279336731098526398390609803239319801398048897
sigma(1310825268269643509279336731098526398390609803239319801398048897^4) = 251 * C251
sigma(251^102) = C245

The factorization is C245 = P69 * P177
P69: 254600736006427372776137727660634479871847016720765139834756753209499
P177: 230458987269072250658153999778689611736692040509480073183678323258029663302691995729286928685305494337086709818726598162179925127930797915136331115104153545784629343508496420647[/code]

[QUOTE=wblipp;229718]Finally the factorization of the roadblock 251^103-1 = P69 * P177. [/QUOTE]

OK. People keep describing "roadblocks". A reasonable question is:
"roadblock to what?"

Please explain the [b]exact[/b] nature of these roadblocks. Also please
explain the result when one of them (such as the one above) is removed.

Is the result simply going to be a raising of the lower bound? If so, what
bound does the group hope to achieve? Is there a definitive list of what
is needed to achieve that goal? Or is this simply an open-ended kind of thing... i.e. keep going until the group gets tired of doing it?
What is the current best lower bound that has been achieved with all
the computation that has been done?

From t400.txt:
sigma(2919196853^18)
r1=464875080800952517348162668359740224950680938541 (pp48)
r2=99334090452543520021712779805845961220011377891059012213242304235588219739882246199209365640091777285341099737 (pp110)

I've also been running ECM against t1200.txt. Results so far:
[code]
sigma(359831030446289824370122802794397026477355920281008605904397865527282755015357815362199168040638501^2)
********** Factor found in step 2: 362840910537001620293543514859
Found probable prime factor of 30 digits: 362840910537001620293543514859
Probable prime cofactor 862021591207283863964797828082453560495288330585474252357901605233343617730010381390689 has 87 digits

sigma(4010350343405246409373637916261249867709^4)
********** Factor found in step 2: 46194603988242790540869409687891711181
Found probable prime factor of 38 digits: 46194603988242790540869409687891711181
Composite cofactor 265032168378429329916853963117382816220476398799676427777094079927737954771967841828723167203840427883811 has 105 digits[/code]

@R D Silverman, according to [url]http://www.oddperfect.org/[/url] the lower bound is now 1250 digits. Look around it for more details.

Chris K

[QUOTE=R.D. Silverman;229732]OK. People keep describing "roadblocks". A reasonable question is: "roadblock to what?"[/QUOTE]

Indeed a reasonable question. The answers are in this thread. A certain notorious member of this forum would send you on your way with a admonition to do your homework, but I'll try to be more helpful.

This thread was started by Joseph Chein requesting help with specific roadblock factorizations for specific lower bounds on Odd Perfect and Odd Multi-Perfect (sigma(N)=aN for other small a) numbers. People enjoyed the activity and asked for more numbers. Joseph was apparently done.

After a few months Pascal Ochem responded to the request with composites from his own work. A bare bones summary of his ongoing work can be seen at[URL="http://www.lri.fr/~ochem/opn/"] his web site[/URL]. There are several new mathematical ideas there, including a method to push bounds on OPNs further with a given set of factorizations and new ways to circumvent road blocks. All of these (at least all thus far described) can fairly be described as refinements on the traditional factor chain method of proof for OPNs.

With the various methods for wringing out higher bounds and working around particular roadblocks, it's hard to tell exactly where the boundaries are right now - I assume that pulling those details together optimally will be part of writing an actual paper at some future time. But the factorizations that block simple standard factor chain proofs tend to be the same factorizations that block the enhanced proofs, although at a higher absolute level and sometimes for different reasons. So we usually describe these factorizations in terms of the roadblocks to the simple standard factor chain proofs.

Many factorizations reported in this thread are not road block factorizations. One of the problems of organizing large scale factoring effort for OPN factor chains is the handling of incomplete factorizations. It is usually possible to extend the chain using any factorization, but it is usually more efficient to extend the chain using the largest factors. This leads to the resource problem of how much effort to put into completing factorizations versus extending with partial factorizations. Factorizations completed later can make large trees of factor chains irrelevant except for occasional reuse. Pascal's various "t" files are lists of first incomplete factorizations - factorizations that we know would never be made irrelevant by other completed factorizations.

I've been thinking it may be time to request a subforum to collect these OPN threads. What do you think?

[QUOTE=wblipp;229744]Indeed a reasonable question. The answers are in this thread. A certain notorious member of this forum would send you on your way with a admonition to do your homework, but I'll try to be more helpful.

[/QUOTE]

I am acutely aware that these factorizations can lead to an increased
lower bound for OPN's. But when one travels a road, one usually has
a destination in mind. I will rephrase: "roadblock to what end?"

[QUOTE]
This thread was started by Joseph Chein requesting help with specific roadblock factorizations for specific lower bounds on Odd Perfect and Odd Multi-Perfect (sigma(N)=aN for other small a) numbers. People enjoyed the activity and asked for more numbers. [/QUOTE]

What is so enjoyable about this particular activity instead of others?

Some of the others (e.g. 17 or Bust) have a well-defined, achievable goal.
Some of these others (e.g. 17 or Bust) will actually lead to the solution
of an open problem. I would not call "raise the bound on OPN's as much as
possible" an open problem, since one can never prove that they don't exist
by doing so.

Other projects (e.g. Cunningham) have the goal of improving algorithms.
Such projects help push the state-of-the-art in those algorithms. The factorizations being done in this thread do not push the art. They are mere
number crunching exercizes.

So let me ask: what is the [b]GOAL[/b] here?

[quote=R.D. Silverman;229745]I am acutely aware that these factorizations can lead to an increased
lower bound for OPN's. But when one travels a road, one usually has
a destination in mind. I will rephrase: "roadblock to what end?"



What is so enjoyable about this particular activity instead of others?

Some of the others (e.g. 17 or Bust) have a well-defined, achievable goal.
Some of these others (e.g. 17 or Bust) will actually lead to the solution
of an open problem. I would not call "raise the bound on OPN's as much as
possible" an open problem, since one can never prove that they don't exist
by doing so.

Other projects (e.g. Cunningham) have the goal of improving algorithms.
Such projects help push the state-of-the-art in those algorithms. The factorizations being done in this thread do not push the art. They are mere
number crunching exercizes.

So let me ask: what is the [B]GOAL[/B] here?[/quote]
Read the bits of the post you didn't quote.
In my opinion they are working towards improving the algorithms to increase the bounds which also means to me that they are improving the chance of them finally putting this problem to bed by proving there are no OPNs. I am sure you know that trying to prove something can help to disprove something.

I answered your new questions. You have now returned to your old questions. I have responded to those before. New details about the new mathematics this is leading to are included in my respose, along with the admission that this is refinement not green field work. I'm not going to participate in a pointless rehash of irreconcilable views.

You (among many others) made this possible - the genie is out of the bottle and people can use it to factor whatever strikes their fancy. I imagine the inventors of television felt a similar frustration at the uses made of their invention - and their frustrations had a similar lack of impact on the users.

Regards,
William