## 4 consecutive gigantic factorizations

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• 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
Message 1 of 4 , Sep 27, 2007
Ken Davis found 28000+ 10043-digit primes for an AP4 search:
http://tech.groups.yahoo.com/group/primeform/message/8762

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
• ... Nicely farmed. ... So you would need only one assist from François for a proven record... David
Message 2 of 4 , Sep 27, 2007
--- In primeform@yahoogroups.com, "Jens Kruse Andersen" <jens.k.a@...>
wrote:

> The expected number of double prp hits was 0.78 which could have
> been improved a little with ECM work if needed.

Nicely farmed.

> 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.

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

David
• ... 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
Message 3 of 4 , Sep 27, 2007
> 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
• ... 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
Message 4 of 4 , Sep 27, 2007
--- 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
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