Wrestling with infinity
- There are an infinite number of primes.
Every prime (in fact every integer) can be written in the form
For k fixed, can it be shown that there exist power series k*2^n+1
with an infinite number of primes?
My hunch is that the answer is yes. There are values of k,
Sierpinski k, where the number of primes is zero, and I think there
are there are series with covering values after a few primes , i.e.
with k=1) and can be easily shown to have a finite number of
My studies of k.2^n+1 seem to suggest that the number of primes at
any given n tend towards a constant, c, where c=p/ln(n). c is
different for different k, and range from 0 in the case of
Sierpinski numbers up to about 13 for the very best Payam k. If ln
(OO) = OO, then p=OO, would seem to be a proposition.
Note: The problem is that my data is only good up to around n=150000.
But I am wrestling with infinity and division, which is a dangerous
thing, I think and I am also dealing with small numbers (<100000
digits in length) and a formula which might be shaky as I can't
determine whether c is constant, and that the formula would show an
ever increasing c or a declining c rather than a limit to which the
formula is tending towards.
Any help from you mathematicians here?