Problems & Puzzles: Problems

Problem 5.- Missed prime factors in Woodall numbers

A Woodall number is of the following form : n*2n -1

The smallest Woodall numbers with unknown prime factors are :

368*2368 -1 = 29*3233254457712259*C97
388*2388 -1=  3*57389*21883688258780797*C98
395*2395 -1=  3*11*587*8467*648624223293259*C99
409*2409 -1 = 11*311*541*49333*5999187334683223*C99

would you like to try to search for those unknown prime factors hidden
in the composites C97, C98, & C99’s ?

***

388*2388 -1 & 409*2409 -1 have been already completely factorised (kindly advised to this pages by Jim Howell, 17/oct/98 & 29/Jan/99). Howell also suggest to add to this problem the following Woodall numbers with unknown prime factors:

W380   has factor  C100
W441   has factor  C100
W542   has factor  C100 (factorised by Jo Yeong Uk, see below)
W370   has factor  C101
W404   has factor  C101

***

Woodall number 441, [441*2^441-1] was previously: W441 = 79 * 881 * 1583 * 40018379959 * 268430213562016447 * C100
Jim Howell (2/99) factorized C100 of W441 using the code GMP-ECM 3, by P. Zimmermann (Inria), 17 Sep 1998, with contributions from T. Granlund, P. Leyland, C. Curry, A. Stuebinger, G. Woltman, JC. Meyrignac., in such a way that that, at the end:

C100 = P36*P65

P36 = 210591719488568010340939210068507793

P65 = 10047327804817920303940778680419610211637795822510390814121375687

***
Jo Yeong Uk (22/6/99) has factored C97 of W368. using the code GMP-ECM 3, by P. Zimmermann (Inria), 17 Sep 1998, with contributions from T. Granlund, P. Leyland, C. Curry, A. Stuebinger, G. Woltman, JC. Meyrignac. Jo Yeong Uk spent about 3 months running a SUN ULTRASPARC 167Mhz. in his department. At the end he got C97 = P40*P57, where:

P40 = 6897087901254198180028594716293059876049 and
P57 = 342123242262565615725400911825586560677842484015014189513
***

At 22/09/99 Paul Leyland wrote:
"The current archive is ftp://ftp.cam.uk.eu.microsoft.com/pub/math/
There you will find that all the Cullen and Woodall numbers with index less
than 400, and all co-factors with under 100 digits have been completed.
(Actually, the latter is not quite true: a C89 was found a couple of weeks
ago but my MPQS run should complete it within a few hours, and yesterday I
found a C80 cofactor which I will finish in the next few days.)

Jim Howell mailed me about his factorization of W_441 and I updated my
tables.  I factored W_368 with MPQS back in November 1998, so the latest
report is that of an independent rediscovery....

At present, the first ten unfactored numbers in each table are:

Index   Woodall
404     107.1937544487.1444695350789.   C101
406     3.25849.376969. C115
411     21617.34214399.13413209791.     C105
412     3.13.823.4909.  C119
413     3.5.1112904944581.      C114
419     3.32363.44048579.       C117
420     23.839.16033.73999.72014309.    C108
423     3537292571.13439189417. C111
433     5.29.   C131
434     5.29.193.12781. C125...
"

***

Jo Yeong Uk (8/10/99) has factored c101 from W404.

"I've factored c101 that divides W404(404*2^404-1)
using GMP-ECM3b.The result is below.

GMP-ECM 3b, by P. Zimmermann (Inria), 1st Dec 1998,
with contributions from T. Granlund, P. Leyland, C. Curry, A. Stuebinger, G.
Woltman, JC. Meyrignac. ... and the invaluable help from P.L. Montgomery.
Input number is
557298504433598372601256881219467879487480495527953239401634749/
11789978853956802558068159264886094463

(101 digits)
Using B1=3000000, B2=300000000, polynomial x^18,
sigma=1472482961
Step 1 took 1876020ms for 39070093 muls, 3 gcdexts
Step 2 took -3468737ms for 15367197 muls, 19566
gcdexts
********** Factor found in step 2:
325095425084328415839823836391348029557
Found probable prime factor of 39 digits:
325095425084328415839823836391348029557
Probable prime cofactor
171426129509216388772807222441109335231265248394904287697565859
has 63 digits
"

***

Jo Yeong Uk (16/12/99) has factored c100 from W542. This is his email:

"I've factored c100 that divides W542(542*2^542-1)
using GMP-ECM4a.The result is below.

GMP-ECM 4a, by P. Zimmermann (Inria), 20 Apr 1999, with contributions from T. Granlund, P. Leyland, C. Curry, A. Stuebinger, G.Woltman, JC. Meyrignac, and the invaluable help from P.L. Montgomery.
Input number is
66585228075377099961244008759818311040665593344454995448018352078424/
10990612613563243845334637541443
(100 digits)
Using B1=3000000, B2=804249600, polynomial x^30, sigma=1557532122,
Step 1 took 1002620ms for 39070093 muls, 3 gcdexts
Step 2 took 900950ms for 18348603 muls, 41103 gcdexts
********** Factor found in step 2:
4046566485214544227712757365757226841987
Found probable prime factor of 40 digits:
4046566485214544227712757365757226841987
Probable prime cofactor
1645474708463781213661725950704321309981224310681219808845889
has 61 digits
"

He also proposes to add the following Woodall numbers to factor:

W668 and W951 both are composite but no known factor.
W511 has C102
W433 has C104
W411 has C105

 Records   |  Conjectures  |  Problems  |  Puzzles