I realized that the series (2^n+1)^2 + (2^n)^2 and (2^n-1)^2 + (2^n)

^2 are nothing but Aurifeuillian Factors.

They have some special properties that I have discovered.

1) Basically these numbers are 2^p+2^((p+1)/2)+1 and 2^p-2^((p+1)/2)+1

2^(2p)+1=(2^p+2^((p+1)/2)+1)*(2^p-2^((p+1)/2)+1)

2^(2p)+1= M2p * L2p (Notation used in literature)

2) p must be prime so that either L or M can be a base 2-PRP.

3) Either L or M is divisible by 5. When p%8=1 or 5 M is divisible by

5 and when p%8=3 or 7 L is divisible by 5. So only one candidate

remains for each prime.

4) So the factors of these numbers are of the form 4*p*k+1.

5) I think their distribution is really similar to mersenne primes,

as most of their properties. I have searched these numbers up to

p=35000 and am continuing to search higher. I have found them to

produce an equal number of primes as mersenne numbers. I think these

primes are the Gaussian equivalents of mersenne primes.

6) I think a top 20 list of these numbers can be started on the

primepages.org web page since these numbers are well known and have

been discussed in a lot of papers.

7) I am not sure if DWT can be used productively, with this series.

But if anyone knows how it can be used productively, please let me

know.

In order to speed up the search to higher n's, I am looking for a

sieve/ Trial factorer. Could someone with the required skill please

write me a program to sieve? I did try myself to write one but it is

not very fast. I am currently using that and sieving all numbers up

to 25G before moving to PRPing. (takes about 30 sec to take a

candidate to 25 G)

Let me know, if any one can help.

-- Harsh Aggarwal

Here are the primes I have found so far.

2^1+2^((1+1)/2)+1

2^1-2^((1+1)/2)+1

- Complete Set -

2^3+2^((3+1)/2)+1

2^3-2^((3+1)/2)+1

- Complete Set -

2^5+2^((5+1)/2)+1

2^7-2^((7+1)/2)+1

2^11+2^((11+1)/2)+1

2^19+2^((19+1)/2)+1

2^29+2^((29+1)/2)+1

2^47-2^((47+1)/2)+1

2^73-2^((73+1)/2)+1

2^79-2^((79+1)/2)+1

2^113-2^((113+1)/2)+1

2^151-2^((151+1)/2)+1

2^157+2^((157+1)/2)+1

2^163+2^((163+1)/2)+1

2^167-2^((167+1)/2)+1

2^239-2^((239+1)/2)+1

2^241-2^((241+1)/2)+1

2^283+2^((283+1)/2)+1

2^353-2^((353+1)/2)+1

2^367-2^((367+1)/2)+1

2^379+2^((379+1)/2)+1

2^457-2^((457+1)/2)+1

2^997+2^((997+1)/2)+1

2^1367-2^((1367+1)/2)+1

2^3041-2^((3041+1)/2)+1

2^10141+2^((10141+1)/2)+1

2^14699+2^((14699+1)/2)+1

2^27529-2^((27529+1)/2)+1

----------------------------------------------------------------------

--- In primenumbers@yahoogroups.com, "eharsh82" <harsh@u...> wrote:

> Here is my proof for infiniteness of these primes

>

> if b=a+1

> then we get 2*a^2+2*a+1 =p

> solving this

> a is an integer if there is a prime p such that 2*p-1=m^2

> or m^2+1/2 is a prime

>

> The distribution of such primes would follow the distribution of

> primes with the formula n^2+1

>

> it is conjectured that such primes are infinite.

>

> ----

> Taken from primepages.com

>

> Are there infinitely many primes of the form n2+1?

> There are infinitely many of the forms n2+m2 and n2+m2+1. A more

> general form of this conjecture is if a, b, c are relatively prime,

a

> is positive, a+b and c are not both even,and b2-4ac is not a

perfect

> square, then there are infinitely many primes an2+bn+c [HW79, p19].

>

> ---

>

>

> What do you all think?

>

> Also the series I talked about, what do you think about's it

> distribution.

>

> a=2^n or 2^n-1

> b=a+1 = 2^n+1 or 2^n

>

> the series reduce to ((2^n-1)^2+1)/2 and ((2^n+1)^2+1)/2

>

> What about the distribution of primes in such a series?

>

> let me know!

>

>

> Harsh Aggarwal

>

>

>

> --- In primenumbers@yahoogroups.com, Andy Swallow

<umistphd2003@y...>

> wrote:

> > On Wed, Feb 04, 2004 at 05:31:24AM -0000, eharsh82 wrote:

> > > I am not sure if this series has finite number of primes or not.

> > > I think it has infinite primes.

> >

> > But that is a question you will never be able to answer, one way

or

> > another, if all you're doing is search for primes of this type

using

> > computer methods. So wouldn't it be more interesting to study the

> > abstract theory? Your original question was about Gaussian

primes,

> or

> > primes congruent to 1 mod 4. That's all interesting and fairly

basic

> > stuff. I would have thought that more informative answers would

be

> found

> > in there.

> >

> > Apologies if I'm talking rubbish. It just seems strange that on

the

> one

> > hand you're interested in whether certain sets contain infinitely

> many

> > primes, yet on the other hand you're studying the sets using

methods

> > guaranteed to not be able to answer the question, :-)

> >

> > Anyway, that's my morning rant out of the way...

> >

> > Andy