--- In firstname.lastname@example.org
> You're trying hard to prove, based on its average gap that
> my average gap is wrong.
Application of the prime number theorem to determine the
average gap between primes in the interval [p[n]^2, p[n+1]^2]
is valid, for large values of the n-th prime p[n], since
the numbers in this range differ little in their relative size.
The average gap is asymptotic to g = log(p[n]) + log(p[n+1])
> Even the preliminary calculations for the interval
> [P_n ^ 2, P_ (n 1) ^ 2] show that the error increases,
> but at certain points it returns to the minimum error.
You have fallen into a trap:
"Hasty generalization, a logical fallacy also known as
'the law of small numbers': the tendency for an initial
segment of data to show some bias that drops out later."
You have blinded yourself by looking only at tiny primes.
Asymptotically, your estimate is hopelessly wrong.
In the range [p[n]^2, p[n+1]^2] between the squares of the
successive primes p[n] = 10^1000 - 1769 and
p[n+1] = 10^1000 + 453 the average gap between primes is
well estimated by the prime number theorem as
g =~ 2000*log(10) =~ 4605.
Your faulty estimate would be smaller, namely
exp(Euler)*1000*log(10) =~ 4101.
Hence you would overestimate the quantity of primes Q,
in this range, by the huge error that I computed for you:
> E =~ 2*exp(-Euler)*Q - Q =~ 1.19*10^999
Please do not blind yourself by looking only at tiny primes.