well as quatratic integers. That is, numbers of the form

|u^p - v^p| with |u - v| = 1 and |z|=|a + b*i|=a^2 + b^2 for guassian

integers or |z|=|a + b*sqrt(N)|=|a^2 - N*b^2| for quadratic integers

(for some integer N that is not a perfect square of course).

Examples:

A/ |(1 + i)^p - 1| which are called Guassian Mersenne Norms. See

http://primes.utm.edu/top20/page.php?id=41,

B/ |(2 + i)^p - (1 + i)^p| which is prime for p=2, 3, 5, 7, 11, 17,

19, 23, ...

C/ |(3 + sqrt(3))^p - 1| = 6^p + 1 - lucasV(6, 6, p) which is prime

for p=2, 3, 11, ..., 151537, 170371 (last 2 exponents are Mike's

records),

D/ |(5 + sqrt(3))^p - 3^p| which is prime for p=2, 7, 13, ...

J-L.

--- In primenumbers@yahoogroups.com, Cletus Emmanuel <cemmanu@y...>

wrote:>

generate primes, I looked at these numbers together with Carol and

>

>

> grostoon <grostoon@y...> wrote: Hi Robert,

>

> --- In primenumbers@yahoogroups.com, "Robert" <rw.smith@b...> wrote:

> >

> > Factors of n^p-(n-1)^p are all of the form F=2kP+1, where k is an

> > integer and P is a prime.

> >

>

> Yes, and if p is a Sophie Germain prime such that p=3 mod 4 and

> (2p+1) divides n-2 then 2p+1 divides n^p-(n-1)^p, which generalizes

> some well known properties of mersenne's numbers factors.

>

> BAck in 1995 when I was trying to find a "super formula" to

Kynea Numbers. I called these numbers Mersenne Class Numbers. I

think that the properties are more or less the same as Mersenne.

I've gone out on a limb to say that people searching for Mersenne

back in the days were aware of the Mersenne Class Numbers, but

thought that the smallest one (Mersenne Numbers) were more

interesting. I do hope that someday we can develop a Mersenne-like

test will be available for these numbers so that we can prove them

prime.>

my mathematics Senior Seminar was 80% about these numbers). I chose

> Although I spent more time on these Classes of numbers (in fact,

to move on with Carol and Kynea Numbers simply because they were

yielded more primes in a given range than these classes of numbers.

You can see how all of these numbers resembles Mersenne Numbers:> Carol Number ---> (2^p-1)^2-2; where p = 2,3,4,...

currently

> Kynea Number ---> (2^p+1)^2-2; where p = 0,1,2,3,4,...

> Mersenne Number-> (2^p-1); where p is prime

> Mersenne Class--> n^p-(n-1)^p; where p is prime

>

> You should take a look at

> http://www.primenumbers.net/prptop/prptop.php for a lot of PRP of

> this form. My current records are 9^170099-8^170099 (162316 digits)

> and 8^173687-7^173687 (156855 digits) and Mike's record is

> 3^256199-2^256199 (122238 digits).

realize that it would go this far.

>

> Impressive records!!!!!

>

> Recently, I found the smallest exponents such that the two "hard

> cases" 48^p-47^p and 61^p-60^p are PRP. They are 58543 and 54517.

>

> This is interesting!!! I knew that 48 was tough, but I didn't

> Now, I'm working on the next hard case 134^p-133^p which I expect

to see a record PRP in this form...

> will give one of the biggest PRP.

>

> Good luck on finding a PRP with that n value. It would be good

>

Number theory

> Regards,

>

> Jean-Louis.

>

>

>

>

>

>

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