Let’s denote by the smallest potential typical primality radius of a given positive integer
.
Writing:
where stands for the number of potential typical primality radii of
below
and
is a quantity depending only on
,
one gets:
and
where is any positive real number less than
.
Hence should be taken as equal to
,
and one would thus get:
Considering the definition of the quantity in the article « Primality radius », the only possible value of
such that:
is:
Therefore one should expect the following relation to hold:
As it can be proven that:
one gets the following average value for :
for some
.
This might help to establish the asymptotic Goldbach’s conjecture (every large enough even integer is the sum of two primes) and Cramer’s conjecture that says:
.