- Ken Davis found 28000+ 10043-digit primes for an AP4 search:

http://tech.groups.yahoo.com/group/primeform/message/8762

He has kindly given me access to 28772 of form k*2^33333+1.

If m and m+1 have known factorization then there are 6 cases

with 2 out of 4 consecutive factorizations:

Start at m-2, m-1, m, 2m-1, 2m, or 3m.

The involved numbers were sieved to 10^12, divided by the found

factors and quickly prp tested with PFGW Version 20050213, using

that they divide an optimized form.

The expected number of double prp hits was 0.78 which could have

been improved a little with ECM work if needed. With a little luck

there was exactly one hit without ECM, for k = 21996007*8.

Let n = 21996007*2^33337. Factorizations:

n = 2^33337*11*29*53*1301

n+1 = 3*5*1129*prp10039

n+2 = 2*(21996007*2^33336+1)

n+3 = 21996007*2^33337+3 (prime)

Trial factoring to 1129 would have been enough.

p = (n+2)/2 was found by Ken and turned out to be a Sophie Germain

prime where 2p+1 = n+3 was easily proven prime with help from p.

This means the factorization only has one prp where two were expected.

My trawl for 3 factorizations with a prp cofactor near a former

top-5000 prime has stopped at 100000 digits.

3 more prp's were found since the last mail:

(2347*2^223281+1)/(3*5*197*1213*1783)

(1363*2^246767+1)/(3*5*7*132*743)

(777*2^247788+1)/(11*754121)

The record now starts with the 74595-digit prime 777*2^247788-1,

found in 2006 by Tony Galvan, NewPGen, Primesearch, LLR.

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

--

Jens Kruse Andersen - --- In primeform@yahoogroups.com, "Jens Kruse Andersen" <jens.k.a@...>

wrote:

> The expected number of double prp hits was 0.78 which could have

Nicely farmed.

> been improved a little with ECM work if needed.

> p = (n+2)/2 was found by Ken and turned out to be a Sophie Germain

So you would need only one assist from François for a proven record...

> prime where 2p+1 = n+3 was easily proven prime with help from p.

David - David Broadhurst wrote:
> So you would need only one assist from François for a proven record...

Yes, his fastECPP would probably take much less time than Ken

used on the 28772 primes, or that would be expected with any

known strategy for 4 gigantic factorizations with prp's allowed.

But a gigantic certification would still be a lot to ask for a record

category I started last month at my own site.

Ken made the hard work for a better known record.

--

Jens Kruse Andersen - --- In primeform@yahoogroups.com, "Jens Kruse Andersen" <jens.k.a@...>

wrote:

> a lot to ask for a record category I started last month

Well you already got someone to make the second progress

on Prouhet-Tarry-Escott in the last 60 years, so you

must be doing something right.

Best regards

David