--- In

primenumbers@yahoogroups.com,

<chitatel2000@...> wrote:

> 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:

http://en.wikipedia.org/wiki/Law_of_small_numbers
"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.

David