Unfurled Mersenne Primes
 

Hi all,
I would like to introduce to you the concept of Unfurled-Mersenne-Primes:

M11@1 = M11 = 2^11 - 1^11 = 2047 is not a prime number.

However the Unfurled-Mersenne-Number:
M11@5 = 313968931
is a prime number.

Let's define Mp@n as

Mp@n = (n+1)^p - n^p

For all primes p and positive integers n.

Generally all prime factors q of Mp@n are such that:
p | (q-1)


Thank you for your time.
:smile::paul:

[QUOTE=a1call;494142]Hi all,
I would like to introduce to you the concept of [STRIKE]Unfurled-Mersenne-Primes[/STRIKE]:
[/QUOTE]
Please don't. Things that already have a name don't need another name.
They are generalized repunit primes (or if you want, uber-generalized repunit primes, that is with fractional b = u/v; and then, in your case - with only u=v+1)

[URL]https://en.wikipedia.org/wiki/Mersenne_prime#Other_generalized_Mersenne_primes[/URL]

Tons of them are known.
[URL="http://www.primenumbers.net/prptop/searchform.php?form=a%5En-b%5En&action=Search"]http://www.primenumbers.net/prptop/searchform.php?form=a%5En-b%5En&action=Search[/URL]

[SPOILER]And of course they only have factors of form 2kp+1, that's bloody obvious.[/SPOILER]

Hi all,
I would like to introduce to you the revised concept of Unfurled-Mersenne-Primes.

Mp@n = (2^(p^(n+1))-1) / (2^(p^n)-1)
For all primes p and positive integers n.
Mp@n may only be prime if Mp is prime.

As an example
M3@1=73

Thank you for your time.:smile:

Cyclotomic primes. What else is new?


Ecclesiastes 1:10

[QUOTE=Batalov;494151]Cyclotomic primes. What else is new?


Ecclesiastes 1:10[/QUOTE]

I'm afraid that stuff as intriguing as they seem are over my head. But thank you for taking the time to try and educate me.:smile:

As for the Bible-Studies, perhaps you have heard of Newfoundland Province in Canada::smile:

[QUOTE]The name "Newfoundland" is a translation of the Portuguese Terra Nova, which literally means "new land" and is also reflected in the French name for the Province's island part (Terre-Neuve). The influence of early Portuguese exploration is also reflected in the name of Labrador, which derives from the surname of the Portuguese navigator João Fernandes Lavrador.[10]

Labrador's name in the Inuttitut language (spoken in Nunatsiavut) is Nunatsuak, meaning "the big land" (a common English nickname for Labrador).[B] Newfoundland's Inuttitut name is Ikkarumikluak meaning "place of many shoals".[/B][/QUOTE]

[url]https://en.wikipedia.org/wiki/Newfoundland_and_Labrador#Etymology[/url]

In any case new or prehistoric as it may be, I can only find the following:

*** 73
*** M3@1 = P2 is Prime.

*** 262657
*** M3@2 = P6 is Prime.

*** 4432676798593
*** M7@1 = P13 is Prime.


Has there been any attempts at primality testing
M521@1
which is a 81556 dd integer?

Thanks again for your time.

[QUOTE=a1call;494154]

Has there been any attempts at primality testing
M521@1
which is a 81556 dd integer?

[/QUOTE]

[CODE]time ./pfgw64 -f -q"(2^(521^(1+1))-1)/(2^521^1-1)"
PFGW Version 3.7.10.64BIT.20150809.x86_Dev [GWNUM 28.7]

(2^(521^(1+1))-1)/(2^521^1-1) has factors: 8143231


real 0m1.743s
user 0m1.724s
sys 0m0.000s
[/CODE]

Cool thanks.:smile:
My Pari code was stuck

On ispseudoprime(a) as well as isprime(a,1) and isprime(a)

Not sure why.

For the record OEIS has been searched in the past and contains a sequence containing the 1st two primes:

[url]https://oeis.org/search?q=73%2C+262657%2C&sort=&language=&go=Search[/url]

But none containing all 3 primes:
[url]https://oeis.org/search?q=73%2C+262657%2C+4432676798593&sort=&language=&go=Search[/url]

There will be only finite number of primes of this sparse form.
One more is (2^59^2-1)/(2^59-1)


[url]http://oeis.org/A156585[/url] : 2, 3, 7, 59

[QUOTE=a1call;494158]Cool thanks.:smile:
My Pari code was stuck

On ispseudoprime(a) as well as isprime(a,1) and isprime(a)

Not sure why.

[/QUOTE]

Obviously, ispseudoprime() does shallow factoring and will take a long while without FFT, but you should have first tried factor(N,10000000) or have written a forprime() loop that bails on the result of a % operation being 0.

[QUOTE=paulunderwood;494161]Obviously, ispseudoprime() does shallow factoring and will take a long while without FFT, but you should have first tried factor(N,10000000) or have written a forprime() loop that bails on the result of a % operation being 0.[/QUOTE]

You are absolutely correct. It's just that I thought all of these would do a trial by division regularly/periodically for small primes rather than loop hopelessly to infinitude in whatever they do.

I think I need a nap now. It's been a busy morning so far:paul:

I will update my code with limits in isprime when I get up.:smile:

[QUOTE=Batalov;494159]There will be only few of the primes of that sparse form.
One more is (2^59^2-1)/(2^59-1)[/QUOTE]

But M59 is not prime.
That ruins the definition as well as my nap.