1. Has anyone worked on the size of gaps between primes and
semiprimes, s, after small primes factors (say p^2 < s < p#^0.5) have
2. If conjecture 53 at
were re-stated as:
If p_(i-1), p_i, p_j, p_k, p_(k+1) are primes such that p_(i-1) < p_i
< p_j < p_k < p_(k+1), and "p_(i-1), p_i" and "p_k, p_(k+1)" are
consecutive prime numbers then there is a
x = floor((p_i*p_j)/p_(i-1)) = p_i + p_j - p_(i-1)
y = floor((p_k*p_(k+1))/p_j) = p_k + p_(k+1) - p_j
in which x_0 = 0, x_1 >=1 and y_0 = 0, y_1 >= 1, and the value of
R_1 = p_j - p_i and
R_2 = p_k - p_j and
m = Pi(p_k) - Pi(p_i)
for every p_j.
Note: I did not state that p_j as consecutive, so a larger p_k and
p_(k+1) or smaller p_(i-1) and p_i could be used as to bring x >0 or
y>0. You may think of R_1 and R_2 as the radius of the zero floor.
What can be said about each of values?
3. Has anyone discovered a relationship between of the x with
Ramanujan prime, R_n, with Chebyshev's bias using Erdos Choquet Theory?
Erdos Choquet Theory, ECT, which proved that there exist at least one
prime of the form 4k+1 and at least one prime of the form 4k+3 between
n and 2n for all n>6.
ECT seems to show only one value per prime of the form while the n
values as with R_n increases to be greater than number, sqrt(R_n), of
Can ECT be made to stair step as R_n does?
Also with the bias, what has been stated for the ratio of (4k + 1) /
(4k -1) relative to the values <= R_n, and for small groupings
(k-tuples) of consecutive prime numbers?