--- In

primeform@yahoogroups.com, "Jens Kruse Andersen"

<jens.k.a@...> wrote:

> Big congratulations on an astonishing record.

I was also astounded. I had been doing some p25 work on a

sample of 1158 (potential) factorizations of 5 consecutive

integers at 500 digits, based on PFGW doublets, obtained by

running with the switches -f5000 -d for the two inner

members of Jarek's (essentially unique?) quartic

construction.

I estimated the Poisson mean, mu(k), for k>8 successive

factorizations, as

mu(k)=(40/25)^2*1158*(k-4)*(25*exp(Euler)/500)^(k-5)

where the first factor signified a willingness to step up to

p40 work if 2 out of k>8 slots had not yet cracked. This

heuristic gave mu(9)=0.93, which suggested that k=9 was

worth trying, without increasing the size of the PFGW

sample. (GMP-ECM is preferable to PFGW, on AMD64s running

linux.)

In the event, my second case with k=8 immediately gave

k=10, with no need for p40, thanks to /two/ flukes

N+0 = 2^2*97*p506

N+1 = 3*19*443*p504

at the lowly p7 level that Pari-GP was using to monitor

possible extensions of sequences, before handing over to

/serious/ GMP-ECM work.

At this point I strongly suspected that there was a

programming bug, since the chance of two free lunches at

p7 level was merely

(7*exp(Euler)/500)^2 = 0.06%

But it was a case of fortune favouring the brave: morally,

I deserved a result at k=9, eventually; happily, I got a

result at k=10, much faster than I expected to crack k=9.

Then there was the residual matter of factorizing 8

algebraic cofactors with 128 digits. (Jens says that this is

"relatively easy", but I do not know how many c128's he has

cracked with MPQS or GNFS. My score is precisely 0, with one

notable disaster trying to parallelize GGNFS on a c124.)

In the event, I did not need Sean Irvines's help: the

hardest factorization in a 128-digit cofactor was merely

c118 = ((13860*(10^60+1898683))^2/8-1)/9106676377 = p45*p73

with

p45 = 612167007455266691724205830229887098156127313

found by Chris Monico's GGNFS, which allows for parallel

sieving and seems reasonably stable below 120 digits.

That factorization took half a day:

> Total sieving time: 61.39 hours. [Shared by 10 processors]

> Total relation processing time: 0.36 hours.

> Matrix solve time: 3.92 hours.

> Time per square root: 0.34 hours.

and was achieved just before the moors become dangerous:

http://en.wikipedia.org/wiki/Glorious_Twelfth
David