Massive P727 found for (10^49081-1)/9-1

AntonVrba · 2005-12-26, 12:51

Folks here is a Xmas present to you

the 727 digit prime factor below is for
(10^(409*3)-1)/9 (which also divides (10^49081-1)/9-1)

12345678901234567890123456789012345678901234567890
56048597739059743500176230768982642214139411802334
99786546561411760392743069141225429028632120282160
30173289675257303884141926548227022154500203647081
89143742433953871265582241598409370945346891683767
04631340795650273383374661026682057679314000813923
24313638565755803466463251993735623526304728365467
13974172221944948815007335847635870892045039295411
13716838038516795713796966562741559751341268792248
21819949032684004058267998143698952156134579716504
66895411432490883165726678515051897532294235172165
60965241583164070851714860767664649082777772946793
84377123048405242912304646833192714476441114216013
58955985908952263848585394777501950185925810161197
27302259015787389757674432851812118641663455398677
120337442484414199381689021

remaining now is a C320 digit composite

18782700121789322140516475898102146084311906005692
31241240418635175653028857901268449933933289884758
08569611799975945808833616874637321320961974341569
00894479486303950957492349113295283059519530039038
15415356665141033911562567345447660481375266829052
84890329795316691965868093340574749208125318555240
06798554161956866521

other prime factors are:

230703986686330645437422372795294965009 ,
7061270715258437 , 22123889761 *) ,
5371851809 , 3334987 *) , 277111 , 1375877 *),
77711 , 13907 , 1637 , 37 , 3

Over 2000 digits of the repunit (10^49081-1)/9-1 are now
in prime factors.

How I found P727, a process of further reducing
(10^(409*3)-1)/9 and Dario Alpern's internet Java
ECM factorisation which found the marked factors *)

happy Xmas
Anton Vrba

akruppa · 2005-12-26, 13:23

How can a 727 digit prime divide a number that has only 409 digits?

Btw, determine all algebraic factors of (10^49081-1)/9-1 first.

Alex

AntonVrba · 2005-12-26, 13:47

Quote:

Originally Posted by akruppa

How can a 727 digit prime divide a number that has only 409 digits?

Btw, determine all algebraic factors of (10^49081-1)/9-1 first.

Alex

read carefully 409*3 = 1227

and test it !

John Renze · 2005-12-26, 15:14

Quote:

Originally Posted by AntonVrba

12345678901234567890123456789012345678901234567890
56048597739059743500176230768982642214139411802334
99786546561411760392743069141225429028632120282160
30173289675257303884141926548227022154500203647081
89143742433953871265582241598409370945346891683767
04631340795650273383374661026682057679314000813923
24313638565755803466463251993735623526304728365467
13974172221944948815007335847635870892045039295411
13716838038516795713796966562741559751341268792248
21819949032684004058267998143698952156134579716504
66895411432490883165726678515051897532294235172165
60965241583164070851714860767664649082777772946793
84377123048405242912304646833192714476441114216013
58955985908952263848585394777501950185925810161197
27302259015787389757674432851812118641663455398677
120337442484414199381689021

This number is not prime, nor does it divide 10^1227-1. See below for the correct factorization. Note also that I reproduced this with Mathematica in about five seconds. This factor has been known for quite some time.

That said, I invite you to continue the effort to prove R49081 prime.

John

In[12]:=
FactorInteger[(10^(3 409)-1)(10^1-1)/((10^409-1)(10^3-1))]//Timing

Out[12]=
{4.641 Second,{{3334987,1},{
22123889761,1},{\
122101609885087549803069193289277079448822848129625671241995756502067340611629\
328187137852419020917651512233903095821505520745954382628708659159770413068456\
864667564702060085057676041295472543648947486382681801929689304732471730931767\
707618738103817640486228998151395492792759871193374877100486754550865865510340\
870101806651347457602337527352535312298806964052663823046221917630683432157634\
612952820816729605471699354623053470575533566822501210347533736672620139077518\
554748493351979776760943613068928131654345095735576869887004368911754285183013\
870391102554258027607900406347346723466371121214609915415510272808900099997625\
057900451426926960318556991496274924157277410187751791255084282826540593649312\
488839736539960894837892356648429319122936042311886045234598893008632717620467\
43350040311309960813,1}}}

AntonVrba · 2005-12-26, 16:49

Sorry I owe you all apology

12345678901234567890123456789012345678901234567890
12210160988508754980306919328927707944882284812962
56712419957565020673406116293281871378524190209176
51512233903095821505520745954382628708659159770413
06845686466756470206008505767604129547254364894748
63826818019296893047324717309317677076187381038176
40486228998151395492792759871193374877100486754550
86586551034087010180665134745760233752735253531229
88069640526638230462219176306834321576346129528208
16729605471699354623053470575533566822501210347533
73667262013907751855474849335197977676094361306892
81316543450957355768698870043689117542851830138703
91102554258027607900406347346723466371121214609915
41551027280890009999762505790045142692696031855699
14962749241572774101877517912550842828265405936493
12488839736539960894837892356648429319122936042311
88604523459889300863271762046743350040311309960813

ia the factor and it has 800 digits

akruppa · 2005-12-26, 17:02

This factorisation is long known and the 800 digit factor has been proven prime some time ago, afaik with ECPP but I don't remember by whom.

See for example http://www.h4.dion.ne.jp/~rep/ for known factors of repunits.

Alex

2005-12-26, 12:51	#1
AntonVrba Jun 2005 2·7² Posts	Massive P727 found for (10^49081-1)/9-1 Folks here is a Xmas present to you the 727 digit prime factor below is for (10^(4093)-1)/9 (which also divides (10^49081-1)/9-1) 12345678901234567890123456789012345678901234567890 56048597739059743500176230768982642214139411802334 99786546561411760392743069141225429028632120282160 30173289675257303884141926548227022154500203647081 89143742433953871265582241598409370945346891683767 04631340795650273383374661026682057679314000813923 24313638565755803466463251993735623526304728365467 13974172221944948815007335847635870892045039295411 13716838038516795713796966562741559751341268792248 21819949032684004058267998143698952156134579716504 66895411432490883165726678515051897532294235172165 60965241583164070851714860767664649082777772946793 84377123048405242912304646833192714476441114216013 58955985908952263848585394777501950185925810161197 27302259015787389757674432851812118641663455398677 120337442484414199381689021 remaining now is a C320 digit composite 18782700121789322140516475898102146084311906005692 31241240418635175653028857901268449933933289884758 08569611799975945808833616874637321320961974341569 00894479486303950957492349113295283059519530039038 15415356665141033911562567345447660481375266829052 84890329795316691965868093340574749208125318555240 06798554161956866521 other prime factors are: 230703986686330645437422372795294965009 , 7061270715258437 , 22123889761 ) , 5371851809 , 3334987 ) , 277111 , 1375877 ), 77711 , 13907 , 1637 , 37 , 3 Over 2000 digits of the repunit (10^49081-1)/9-1 are now in prime factors. How I found P727, a process of further reducing (10^(4093)-1)/9 and Dario Alpern's internet Java ECM factorisation which found the marked factors ) happy Xmas Anton Vrba

2005-12-26, 16:49	#5
AntonVrba Jun 2005 2·7² Posts	CORRECTION Sorry I owe you all apology 12345678901234567890123456789012345678901234567890 12210160988508754980306919328927707944882284812962 56712419957565020673406116293281871378524190209176 51512233903095821505520745954382628708659159770413 06845686466756470206008505767604129547254364894748 63826818019296893047324717309317677076187381038176 40486228998151395492792759871193374877100486754550 86586551034087010180665134745760233752735253531229 88069640526638230462219176306834321576346129528208 16729605471699354623053470575533566822501210347533 73667262013907751855474849335197977676094361306892 81316543450957355768698870043689117542851830138703 91102554258027607900406347346723466371121214609915 41551027280890009999762505790045142692696031855699 14962749241572774101877517912550842828265405936493 12488839736539960894837892356648429319122936042311 88604523459889300863271762046743350040311309960813 ia the factor and it has 800 digits

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2005-12-26, 13:23	#2
akruppa "Nancy" Aug 2002 Alexandria 2,467 Posts	How can a 727 digit prime divide a number that has only 409 digits? Btw, determine all algebraic factors of (10^49081-1)/9-1 first. Alex

2005-12-26, 17:02	#6
akruppa "Nancy" Aug 2002 Alexandria 2,467 Posts	This factorisation is long known and the 800 digit factor has been proven prime some time ago, afaik with ECPP but I don't remember by whom. See for example http://www.h4.dion.ne.jp/~rep/ for known factors of repunits. Alex