Collated by Paul Zimmermann | ## Record Factors Found By The p+1 Method |
---|

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 | L_{2366} | A. Kruppa P. Montgomery | 31.10.2007 (*) | 10^{11} | 10^{15} |

55 | 1273305908528677655311178780176836847652381142062038547 | 782*6^{782}+1 | P. Leyland | 16.05.2011 | 10^{9} | 10^{13} |

53 | 60120920503954047277077441080303862302926649855338567 | 682 * 5^{682} - 1 | P. Leyland | 26.02.2011 | 10^{8} | 10^{12} |

52 | 7986478866035822988220162978874631335274957495008401 | 47^{146}+1 | P. Montgomery | 22.07.2007 (*) | 10^{10} | 10^{12} |

51 | 304494037912467064027747619058145093270322935609211 | 6571^{41}-1 | A. Reich | 24.04.2012 | 10^{7} | 10^{12} |

49 | 1809864641442542950172698003347770061601055783363 | L_{2442} | A. Kruppa | 25.12.2007 | 10^{8} | 10^{14} |

49 | 1579932223057448311905111561701925358935050104819 | 75^{142}+142^{75} | O. Ã–stlin | 09.08.2013 | 10^{9} | 10^{15} |

48 | 884764954216571039925598516362554326397028807829 | L_{1849} | A. Kruppa | 29.03.2003 (*) | 10^{8} | 10^{10} |

48 | 544759909571610453564318387860510712113049589379 | 608 * 11^{608} - 1 | P. Leyland | 05.05.2011 | 10^{9} | 10^{11} |

48 | 495777646717226682946854701466525839651092192469 | L(4259) | N. Daminelli | 07.09.2009 | 10^{8} | 10^{12} |

Digits | p | Factor Of | Found By | Year | B1 | B2 |
---|---|---|---|---|---|---|

39 | 134368561962115712052394154476370507609 | 162*11^{162}+1 | P. Leyland | 2002 (*) | 10^{7} | 10^{9} |

37 | 4190453151940208656715582382315221647 | 45^{123}+1 | P. Montgomery | 1994 (*) | 10^{7} | 10^{9} |

25 | 6563589514883537474323387 | L_{442} | P. Montgomery | 1985 (*) | 10^{6} | 10^{7} |

21 | 122551752733003055543 | 2^{439}-1 | R. Brent | 1981 (*) | 10^{5} | 10^{6} |

18 | 347366417511089201 | L_{254} | H. Williams | 1981 (*) | 10^{5} | 10^{6} |

- L
_{n}is the n^{th}number in the Lucas sequence, F_{n}is the n^{th}number in the Fibonacci sequence.