## Big PrimoProth prime number

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• 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
Message 1 of 5 , Oct 28, 2008
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
• ... A Proth prime is k*2^n+1 with k
Message 2 of 5 , Oct 28, 2008
Didier Boivin wrote:
> 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)
>
> 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 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 primorial:
m#*2^n+1 with m# < 2^n
However, m#*2^n-1 with m# < 2^n also appears to be allowed.
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
• ... After looking at more Google hits, it actually appears that the few people searching or listing PrimoProths use different definitions or don t state a
Message 3 of 5 , Oct 28, 2008
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 primorial:
> m#*2^n+1 with m# < 2^n
> 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 people
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
• ... people ... state a ... I was involved some years ago as a coordinator for primes of the series n#/2*2^n+/1, which I called primoproths, influenced by Henri
Message 4 of 5 , Oct 29, 2008
--- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
<jens.k.a@...> wrote:
>
> 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
primorial:
> > m#*2^n+1 with m# < 2^n
> > 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
people
> 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 Andersens
>

I was involved some years ago as a coordinator for primes of the
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:
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
• ... Hello Jens and Robert, Thanks for your valuable comments. I forgot to check the criteria of m#
Message 5 of 5 , Nov 6, 2008
--- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
<jens.k.a@...> wrote:
>

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

Hello Jens and Robert,
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
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