- The series k^2+(k+1)^2 and k^4+(k+1)^4, k from 1 to infinity produce may primes (prps), as factors of these series (and all series of form k^((2^n))+(k+1)^((2^n))appear to be similar to those factors in cyclotomic polynomials.

n=1 and 2 appear very rich, as the formula produces quite small primes and prps, but counting primes (prps) for the series k from 1 to k(x) it appear that n=2 has more than n=1, for low x.

What is quite fascinating is if we look at the mod 2 values of k that produce prps for n=1,2 and race k=0mod2 against k=1mod2. I looked up to k=1200000 and over 100000 of these are prp, for both n=1 and 2, and the even values of k were in the lead for a goodly proportion of the time, in fact for 99%+. At k=1200000, n=2 there were approx 600 more evens than odds.

It would be interesting to know if this is a permanent phenomenon or one that will eventually reverse. My guess that it will reverse and that for some value of k>1200000, there are more odds than evens.

Sorry if this area is well researched, but I always find google is not so good at looking for mathematical polynomial equations, unless they have a name.

Regards

Robert Smith - --- In primenumbers@yahoogroups.com,

"Robert" <robert_smith44@...> wrote:

> over 100000 of these are prp

...

> there were approx 600 more evens than odds

Since

600/sqrt(100000) < 2

this seems to tell us little about what might happen later.

David - --- In primenumbers@yahoogroups.com, "Robert" <robert_smith44@...> wrote:
>

After a good deal of lead swapping at small k, even numbers predominate from k=44886 until k=4245827

> The series k^2+(k+1)^2 and k^4+(k+1)^4, k from 1 to infinity produce may primes (prps), as factors of these series (and all series of form k^((2^n))+(k+1)^((2^n))appear to be similar to those factors in cyclotomic polynomials.

>

> n=1 and 2 appear very rich, as the formula produces quite small primes and prps, but counting primes (prps) for the series k from 1 to k(x) it appear that n=2 has more than n=1, for low x.

>

> What is quite fascinating is if we look at the mod 2 values of k that produce prps for n=1,2 and race k=0mod2 against k=1mod2. I looked up to k=1200000 and over 100000 of these are prp, for both n=1 and 2, and the even values of k were in the lead for a goodly proportion of the time, in fact for 99%+. At k=1200000, n=2 there were approx 600 more evens than odds.

>

>

Regards

Robert Smith