- Hello,

I'm glad to announce the discovery of two big (most probably the

largest known) primoproth prime numbers.

As a reminder, these number can be written like m#*2n +/- 1, m# being

the primorial of the prime m.

155161#*2^2405+1 is prime (67961 digits)

155167#*2^2405+1 is prime (67966 digits)

The whole job (PRP test and primality proof) has been done with

winPFGW (1.29), by keeping the exponant constant and increasing the

value of m.

During this search, other number have been proven primes :

17#*2^2405+1

947#*2^2405+1

977#*2^2405+1

2969#*2^2405+1

31907#*2^2405+1

To my knowledge, these 2 records cannot be registered in the Prime

Pages, but at least, it is shared with all of you.

Best regards.

Didier Boivin - Didier Boivin wrote:
> I'm glad to announce the discovery of two big (most probably the

A Proth prime is k*2^n+1 with k < 2^n.

> largest known) primoproth prime numbers.

> As a reminder, these number can be written like m#*2n +/- 1, m# being

> the primorial of the prime m.

> 155161#*2^2405+1 is prime (67961 digits)

> 155167#*2^2405+1 is prime (67966 digits)

>

> During this search, other number have been proven primes :

> 17#*2^2405+1

> 947#*2^2405+1

> 977#*2^2405+1

> 2969#*2^2405+1

> 31907#*2^2405+1

>

> To my knowledge, these 2 records cannot be registered in the Prime

> Pages, but at least, it is shared with all of you.

A PrimoProth prime is based on this, a Proth prime where k is a primorial:

m#*2^n+1 with m# < 2^n

However, m#*2^n-1 with m# < 2^n also appears to be allowed.

See http://www.primenumbers.net/Henri/us/FactPrimus.htm

The only PrimoProths in your post are those with 17#, 947#, 977#.

If the smallest primorials are allowed (and I haven't seen a rule excluding

them) then many of the largest known primes can actually be written as

PrimoProths, for example all of the forms:

2^n-1 =1#*2^n-1

3*2^n+/-1 = 3#*2^(n-1)+/-1

15*2^n+/-1 = 5#*2^(n-1)+/-1

If the Prime Pages had a top-20 for PrimoProths without tiny primorials then

I guess more people would search them and the top-20 limit would be above

the 5000th prime so they would have qualified for the database anyway.

--

Jens Kruse Andersen - I wrote:
> A Proth prime is k*2^n+1 with k < 2^n.

After looking at more Google hits, it actually appears that the few people

> A PrimoProth prime is based on this, a Proth prime where k is a primorial:

> m#*2^n+1 with m# < 2^n

> However, m#*2^n-1 with m# < 2^n also appears to be allowed.

searching or listing PrimoProths use different definitions or don't state a

definition at all. This confusion may be caused by Henri Lifchitz who

introduced the concept but himself seems unclear with the definiton and

inconsistent in the alleged records, both regarding whether there is an

upper and a lower limit on m#.

--

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

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

primorial:

> I wrote:

> > A Proth prime is k*2^n+1 with k < 2^n.

> > A PrimoProth prime is based on this, a Proth prime where k is a

> > m#*2^n+1 with m# < 2^n

people

> > However, m#*2^n-1 with m# < 2^n also appears to be allowed.

>

> After looking at more Google hits, it actually appears that the few

> searching or listing PrimoProths use different definitions or don't

state a

> definition at all. This confusion may be caused by Henri Lifchitz who

I was involved some years ago as a coordinator for primes of the

> introduced the concept but himself seems unclear with the definiton and

> inconsistent in the alleged records, both regarding whether there is an

> upper and a lower limit on m#.

>

> --

> Jens Kruse Andersens

>

series n#/2*2^n+/1, which I called primoproths, influenced by Henri

-see http://home.btclick.com/rwsmith/pp/page1.htm

As suspected the largest primes discovered involve k=3 and k=15. The

Riesel series k=15 is being actively searched and is at n=1834000 see:

http://www.mersenneforum.org/showthread.php?t=6338. k=3 search is well

known.

The attractions of primoproths were diminished when it was discovered

that Payam numbers gave greater density of primes and active searching

went into abeyance.

But these are good finds, so congrats - --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"

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

primorial:

> A PrimoProth prime is based on this, a Proth prime where k is a

> m#*2^n+1 with m# < 2^n

Hello Jens and Robert,

> However, m#*2^n-1 with m# < 2^n also appears to be allowed.

> See http://www.primenumbers.net/Henri/us/FactPrimus.htm

> The only PrimoProths in your post are those with 17#, 947#, 977#.

>

Thanks for your valuable comments. I forgot to check the criteria of

m# < 2^n, mainly because it was a search with constant n.

Robert points out that these numbers are not records at all.

Surprinsingly, the site has not been updated since 2002. Maybe due to

lack of time, or no more interest or contributors.

BR