## 1 Bernoulli and 5 An factorisations

```    Subject: 1 Bernoulli and 5 An factorisations
Date: Mon, 02 Nov 1998 10:12:34 +0000
From: Allan MacLeod <MACL-MS0@wpmail.paisley.ac.uk>
To: kc2h-msm@asahi-net.or.jp

Hello,

The following results appeared during the weekend.

Allan MacLeod

(a) The factorisation of the remaining composite of Bernoulli 146
into  P32 * P34 * P34

Input number is
269116590632726862241599354298826948271748399014997974494963599274965614871988049574489970791060991 (99 digits)

Using B1=3000000, B2=300000000, polynomial x^30, sigma=630597632
Step 1 took 672912ms for 39102834 muls, 3 gcdexts
Step 2 took 319175ms for 16480797 muls, 29004 gcdexts
********** Factor found in step 2: 37996324998547740539691528067877
Found probable prime factor of 32 digits: 37996324998547740539691528067877
Composite cofactor 7082700514931714608424383737254089922262631966445989054616137913683 has 67 digits

Input number is
7082700514931714608424383737254089922262631966445989054616137913683 (67 digits)

Using B1=3000000, B2=300000000, polynomial x^30, sigma=749175733
Step 1 took 337857ms for 39102834 muls, 3 gcdexts
Step 2 took 164945ms for 16480797 muls, 29004 gcdexts

********** Factor found in step 2: 4036138055144761320534304068715607
Found probable prime factor of 34 digits: 4036138055144761320534304068715607
Probable prime cofactor 1754821172656266926966923716442469 has 34 digits

(b) Factor of A70 = 70! - 69! + 68! - ....

Input number is
859078741381485846771480781396806231586766814640923689834377870700001999521248398187 (84 digits)
Using B1=3000000, B2=300000000, polynomial x^30, sigma=1822271789
Step 1 took 617197ms for 39102834 muls, 3 gcdexts
Step 2 took 239451ms for 13394950 muls, 26969 gcdexts

********** Factor found in step 2: 585030093394485946687838023247

Found probable prime factor of 30 digits: 585030093394485946687838023247

Probable prime cofactor 1468435130228777520806101110144198014434899612807530021 has 55 digits

(c) Factor of A77

Input number is
423924430121577093663604665618738428866938776566665245999701885276513297515199 (78 digits)
Using B1=3000000, B2=300000000, polynomial x^30, sigma=972015242
Step 1 took 610000ms for 39102834 muls, 3 gcdexts

********** Factor found in step 1: 17591036707326253625414788854031670107

Found probable prime factor of 38 digits: 17591036707326253625414788854031670107

Probable prime cofactor 24098888381321069342745720538599406741357 has 41 digits

(d) Factors of A82 into P31 * P29 * P43

Input number is
62937272289325306279305300003450030931627644452080234239146224241178226225750471622048311877940966703 (101 digits)

Using B1=3000000, B2=300000000, polynomial x^30, sigma=1620692091
Step 1 took 669175ms for 39102834 muls, 3 gcdexts
Step 2 took 316759ms for 16480797 muls, 29004 gcdexts

********** Factor found in step 2: 1361843985381500874962939678969
Found probable prime factor of 31 digits: 1361843985381500874962939678969

Composite cofactor 46214744834882345710554126266356144933371170136427326716235875870435687 has 71
digits

Input number is
46214744834882345710554126266356144933371170136427326716235875870435687 (71 digits)
Using B1=3000000, B2=300000000, polynomial x^30, sigma=2140987371
Step 1 took 416813ms for 39102834 muls, 3 gcdexts
Step 2 took 200274ms for 16480797 muls, 29004 gcdexts

********** Factor found in step 2: 31253997184032546487207810861

Found probable prime factor of 29 digits: 31253997184032546487207810861

Probable prime cofactor 1478682696576588383655163289071530063028067 has 43 digits

(e) Factors of A90

Input number is
49877967180978338576246594784518191787007540514926385103788159802480881731335303745593600990648267904551 (104 digits)
Using B1=3000000, B2=300000000, polynomial x^30, sigma=1754054057
Step 1 took 671043ms for 39102834 muls, 3 gcdexts
Step 2 took 323077ms for 16480797 muls, 29004 gcdexts

********** Factor found in step 2: 55019405240237536315808689741

Found probable prime factor of 29 digits: 55019405240237536315808689741

Probable prime cofactor 906552278476847444160580153565664347318791317572967678202253771246791481411
has 75 digits

(f) Factors of A94

Input number is
107745253847603392494772654100857391663112957777817924584621505847263825176047393003652310766916363930287761382063980793761156049285396710361 (141 digits)
Using B1=3000000, B2=300000000, polynomial x^30, sigma=491176053
Step 1 took 1402142ms for 39102834 muls, 3 gcdexts
Step 2 took 670165ms for 16480797 muls, 29004 gcdexts

********** Factor found in step 2: 16961617771107656498762423

Found probable prime factor of 26 digits: 16961617771107656498762423

Probable prime cofactor
6352298188863574899660125160672761857177312600750424600022590896869845596544818074214778062973059728996047576845807 has 115 digits
```

E-mail : kc2h-msm@asahi-net.or.jp
Hisanori Mishima