The Fischbach Prime a mersenne variation
 

Iv'e come up with a prime number generator that seems to generate
large primes, as far as I can test with a slightly greater effeciency than
the mersenne equation can.


Here's the equation
(2^prime +1)/3= possibly prime



Here's a side by side test between the Fischbach prime and the mersennne prime


The mersenne prime is prime when 2 is raised to the following
powers, 2,3,5,7,13,17,19,31,or 61.


The Fischbach prime is prime when 2 is raised to the following
powers,3,5,7,11,13,17,19,23,31,or 43.



You people seem to be so clever on this forum why don't you
come up with a system to test the Fischbach prime for primality
and generate a prize winning prime and take the money to the bank.

[QUOTE=Carl Fischbach;185879]Iv'e come up with a prime number generator that seems to generate
large primes, as far as I can test with a slightly greater effeciency than
the mersenne equation can.


Here's the equation
(2^prime +1)/3= possibly prime



Here's a side by side test between the Fischbach prime and the mersennne prime


The mersenne prime is prime when 2 is raised to the following
powers, 2,3,5,7,13,17,19,31,or 61.


The Fischbach prime is prime when 2 is raised to the following
powers,3,5,7,11,13,17,19,23,31,or 43.



You people seem to be so clever on this forum why don't you
come up with a system to test the Fischbach prime for primality
and generate a prize winning prime and take the money to the bank.[/QUOTE]Wow, so for all numbers n<=61, countof(Mersenne primes)=9 and countof(Fischbach primes)=10.

And that is significant because ... ?

What about for n<=1million? What is the ratio?

Is a "Fischbach prime" test more (or even as) efficient as a Mersenne prime test? If not and the test takes 10 times longer then can we expect a corresponding 10+ times increase in the number of primes found?

Stop calling [URL="http://en.wikipedia.org/wiki/Wagstaff_prime"]primes named after other people[/URL] by your own name!

The mersenne prime has shortcuts for testing primality I looked at my
equation for shortcuts for primality testing and couldn't find any, someone else may have some better luck, it's a lucrative proposition
if you find them.

[QUOTE=axn;185892]Stop calling [URL="http://en.wikipedia.org/wiki/Wagstaff_prime"]primes named after other people[/URL] by your own name![/QUOTE]

ECPP is the fastest test for proving the primality of the Wagstaff ("Fischbach") prime (according to Wikipedia). That's a run-time of O(log^5 n) right? The LL test (with FFT) has a run-time of O(log^2 n log log n) when testing the primality of a Mersenne number n? Of course, I might/will be wrong with some of this information since I am not an expert in compuational number theory.
So I doubt anyone can come up with a primality test to prove the primality of Wagstaff ("Fischbach") primes that cuts the run-time by a poly-logarithmic factor. Of course, a probable primality test may suffice. Miller-Rabin has a run-time of [TEX]\tilde{O}(\log^2 n)[/TEX] with a probability of error at most of 1/4 per iteration. Grantham's RQFT cuts this probability of error down to at most 1/7710 per iteration. Zhang's One Parameter Quadratic Base Test (a variant of the BPSW test) runs in about the same time (see his paper "A One Parameter Quadratic-Base Version of the Baillie-PSW Probable Prime Test") and has a much less probability of error than RQFT.

A [B][U]TRIVIAL[/U][/B] probable prime test which has a much lower probability of error than either RQFT and OPQBT that runs in [TEX]\tilde{O}(k \log^2 n)[/TEX] time, where k=7 in the worst possible case, is by executing 1 iteration in RQFT followed by 1 iteration in OPQBT. I have attached a pdf file describing this RQFT+OPQBT test (I wrote it a couple months ago just for fun). It also talks more about Zhang's OPQBT in a bit more detail:
[ATTACH]3991[/ATTACH]
P.S. I use the term "Selfridge" in this pdf, as it is a term coined and popularized by Jon Grantham which refers to the run-time of the Miller-Rabin test (in other words, [TEX]\tilde{O}(\log^2 n)[/TEX]).
[B][U]Comments are more than welcome regarding the pdf file I attached[/U][/B].

Aww, you guys all gave the game away so early. :sad:

I wanted Carl Fischbach to see if he could find the Wagstaff primes below 1million by himself, hence my question about the ratio for n<=1million. :mad:

I'm not making any claims beyond what Iv'e tested, the proof you want
is beyond my scope.

[QUOTE=Carl Fischbach;185903]I'm not making any claims beyond what Iv'e tested, the proof you want
is beyond my scope.[/QUOTE]
Carl Fischbach,

Please look at my post; it is meant to be of help to you (I hope so at least): [url]http://mersenneforum.org/showpost.php?p=185894&postcount=5[/url]

Kind Regards,
flouran

Iv'e reinvented the wheel again, I was totally unfamiliar with the wagstaff
prime I'm half asleep here.

Where is Dr. Bob when you need him?

other primes (2^n+1)/3 for n=
61, 79, 101, 127, 167, 191, 199, 313, 347 (limit n=500)

see [url=http://factordb.com/search.php?query=%282%5Ex%2B1%29%2F3&v=x&x=1&E=1&Prp=1&P=1&C=1&CF=1&of=H&pp=500&sw=Update]here[/url]