Collated by
Paul Zimmermann

## Record Factors Found By The p+1 Method

[Pollard's p-1] [Lenstra's ECM]

This method of integer factorisation was described by H. C. Williams in "A p+1 Method of Factoring" Mathematics of Computation 39 (159), 1982.

It can find a large factor p very quickly when p+1 is composed of small factors.

This table lists the 10 largest factors found by this method, of which I am aware (lines with an asterix design factors that were at one time the current record). If you know of any others, please email me at zimmerma at loria.fr. Note that the values of B1 and B2 shown here are the minimum power of ten needed to have found the factor and not necessarily the ones actually used by the finder.

Digits p Factor Of Found By Year B1 B2
60 725516237739635905037132916171116034279215026146021770250523 L2366 A. Kruppa
P. Montgomery
31.10.2007 (*) 1011 1015
55 1273305908528677655311178780176836847652381142062038547 782*6782+1 P. Leyland 16.05.2011 109 1013
53 60120920503954047277077441080303862302926649855338567 682 * 5682 - 1 P. Leyland 26.02.2011 108 1012
52 7986478866035822988220162978874631335274957495008401 47146+1 P. Montgomery 22.07.2007 (*) 1010 1012
51 304494037912467064027747619058145093270322935609211 657141-1 A. Reich 24.04.2012 107 1012
49 1809864641442542950172698003347770061601055783363 L2442 A. Kruppa 25.12.2007 108 1014
49 1579932223057448311905111561701925358935050104819 75142+14275 O. Ã–stlin 09.08.2013 109 1015
48 884764954216571039925598516362554326397028807829 L1849 A. Kruppa 29.03.2003 (*) 108 1010
48 544759909571610453564318387860510712113049589379 608 * 11608 - 1 P. Leyland 05.05.2011 109 1011
48 495777646717226682946854701466525839651092192469 L(4259) N. Daminelli 07.09.2009 108 1012

#### Previous records

We list here only the factors that were first at a given time (to our best knowledge), for historical interest.
Digits p Factor Of Found By Year B1 B2
39 134368561962115712052394154476370507609 162*11162+1 P. Leyland 2002 (*) 107 109
37 4190453151940208656715582382315221647 45123+1 P. Montgomery 1994 (*) 107 109
25 6563589514883537474323387 L442 P. Montgomery 1985 (*) 106 107
21 122551752733003055543 2439-1 R. Brent 1981 (*) 105 106
18 347366417511089201 L254 H. Williams 1981 (*) 105 106

#### Notes

1. Ln is the nth number in the Lucas sequence, Fn is the nth number in the Fibonacci sequence.