--- In

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

>

> 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.

>

>

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

Regards

Robert Smith