- Let n = 378149751*2^27186-2

n = 2*(378149751*2^27185-1) (2*prime)

n+1 = 378149751*2^27186-1 (prime)

n+2 = 378149751*2^27186 = 2^27186*3^2*7*17*353081

n+3 = 378149751*2^27186+1 = 5*1019*24923*43651*989239*P8174

The 8174-digit cofactor of n+3 has been proven prime through a joint effort of Geoffrey Hird and myself, and makes the Top-20 for ECPP proofs:

http://primes.utm.edu/primes/page.php?id=92563

This prime is currently the fifth-largest candidate certified by Primo, with a certificate available at:

http://www.ellipsa.eu/public/primo/files/ecpp8174.zip

The two primes 378149751*2^27185-1 and 378149751*2^27186-1 are a Sophie Germain byproduct of a recent CC3 (1st kind) search.

n-1 = 378149751*2^27186-3 = 3*53*127*7043*626921*24117041920337*C8166

n+4 = 378149751*2^27186+2 = 2*11*561139037952529*C8177

n-1 and n+4 have been tested with 74 ECM curves at B1=11000.

Tom - Tom wrote:
> Let n = 378149751*2^27186-2

Congratulations!

>

> n = 2*(378149751*2^27185-1) (2*prime)

> n+1 = 378149751*2^27186-1 (prime)

> n+2 = 378149751*2^27186 = 2^27186*3^2*7*17*353081

> n+3 = 378149751*2^27186+1 = 5*1019*24923*43651*989239*P8174

>

> The 8174-digit cofactor of n+3 has been proven prime through a joint

> effort of Geoffrey Hird and myself, and makes the Top-20 for ECPP proofs

http://users.cybercity.dk/~dsl522332/math/consecutive_factorizations.htm

is updated with a note that the certificate is being verified. Marcel Martin

lists it on the Primo Top-20 and I assume there will be no problems.

I'm pleasantly surprised that somebody has run Primo at 8174 digits for

my record page.

--

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

"tjw99" <tjw99@...> wrote:>

When

> Let n = 378149751*2^27186-2

>

> n = 2*(378149751*2^27185-1) (2*prime)

> n+1 = 378149751*2^27186-1 (prime)

> n+2 = 378149751*2^27186 = 2^27186*3^2*7*17*353081

> n+3 = 378149751*2^27186+1 = 5*1019*24923*43651*989239*P8174

http://primes.utm.edu/primes/page.php?id=92563

was posted, by Geoffrey, I wondered:

what cunning problem might this prime solve?

Thanks, Tom, for your denouement, now added as a comment.

David - David wrote:
> When

When I saw Tom Wu in the prover code I felt certain about the problem it

> http://primes.utm.edu/primes/page.php?id=92563

> was posted, by Geoffrey, I wondered:

> what cunning problem might this prime solve?

solved and only wondered about a little detail.

A few tests confirmed my assumption about the problem and I posted the below

before the record had been announced or submitted.

----- Original Message -----

From: "Jens Kruse Andersen" <jens.k.a@...>

To: "tjw99" <tjw99@...>

Sent: Monday, May 03, 2010 2:27 AM

Subject: Proven factorization of 4 consecutive 8193-digit numbers

> Congratulations on http://primes.utm.edu/primes/page.php?id=92563

> I'm honored that somebody would run Primo at 8174 digits for my record page.

>

> I'm a little curious why it was submitted as

> (378149751*2^27186+1)/(989239*5542921182935)

>

> I would have expected either

> (378149751*2^27186+1)/5483273808085436465 or

> (378149751*2^27186+1)/(5*1019*24923*43651*989239)

> but it's not important for the Prime Pages.

>

> --

> Jens Kruse Andersen

It turned out the latter form was submitted but the Prime Pages

canonicalization converted it.

--

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

"Jens Kruse Andersen" <jens.k.a@...> wrote:

> When I saw Tom Wu in the prover code I felt certain about

thus confirming that Jens is quicker on the uptake than am I.

> the problem it solved

However, my slower brain recently had a (perhaps) neat

idea that may (or may not) result in another update for

Jens to perform, perhaps not before too long.

Festina lente!

David