- Let

m = 67440294559676054016000

a = (y-22^2)*(y-61^2)*(y-86^2)*(y-127^2)*(y-140^2)*(y-151^2)/(6*m)

b = (y-35^2)*(y-47^2)*(y-94^2)*(y-121^2)*(y-146^2)*(y-148^2)/m

Then b = 6*a+1. Moreover, with

y = (m*(10^96+9581328)+22)^2

each element of

[3*a-1, 6*a-1, (3*a+1)/(23*24943*13071241), 2*a+1]

is prime.

By factorizing all 24 of the 119-digit algebraic cofactors

in a and b, we obtain complete factorizations of the 6

successive 1404-digit integers 6*a+k-2 with k in [0,5].

Proof:

http://physics.open.ac.uk/~dbroadhu/cert/ifac6a.zip

Software used:

GGNFS

GMP-ECM

Msieve

OpenPFGW

Pari-GP

Primo

Neighbouring composites:

c1376=(2*a-1)/(59*104882036089*123707274359671);

c1403=(3*a+2)/2;

subjected to 320 ECM curves at B1=250000

David Broadhurst

20 August 2007 - David Broadhurst wrote:
> By factorizing all 24 of the 119-digit algebraic cofactors

Congratulations!

> in a and b, we obtain complete factorizations of the 6

> successive 1404-digit integers 6*a+k-2 with k in [0,5].

http://hjem.get2net.dk/jka/math/consecutive_factorizations.htm is updated.

The algebraic cofactors have been upgraded from

"relatively easy to factor" to "possible to factor"

since your 1104-digit record for 6 numbers.

Donovan Johnson has set a 2135-digit record for 5 numbers.

It contains a twin prime but it and others were computed to

set this record, so it doesn't count as a record with an "origin".

--

Jens Kruse Andersen - --- In primeform@yahoogroups.com, "Jens Kruse Andersen"

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

> Congratulations!

updated.

> http://hjem.get2net.dk/jka/math/consecutive_factorizations.htm is

I liked the peculiarity that 3/4 of the 1400-digit primes

were provable by BLS. Thay was not my intent;

it just happened that way: the 3-constellation

[3*a-1, 6*a-1, 2*a+1] fell out of pfgw -f -d,

unbidden.

In the long term, the k=5 record will need

a 4-constellation, to avoid ECPP. But I leave

that to the master sievers of linear forms.

I prefer shallow sieving of 12th (resp. 8th)

order polynomials for k = 6,7 (resp. 8,9,10),

followed by serious ECM in a minority of

promising cases.

I chose 1400 digits for k=6 so that any

significant later improvement would entail SNFS,

rather than GNFS, which works fine for me

below 120 digits.

David