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• ... P_n =~ 98804779450787677801, by Mertens theorem David
Message 1 of 4 , Dec 3 12:46 AM
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<chitatel2000@...> wrote:

> Calculate (P_n) at 82

P_n =~ 98804779450787677801, by Mertens' theorem

David
• Hello David. You do not understand the problem, hence your error. Need from the formulas for calculating the number of prime numbers, go to the formulas of the
Message 2 of 4 , Dec 9 10:58 PM
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Hello David.
You do not understand the problem, hence your error.
Need from the formulas for calculating the number of prime numbers, go to the formulas of the average gap between primes. And then:
You can not know what formula is the average gap for the interval [0, P_n ^ 2] does not work for the interval [P_n ^ 2, P_ (n 1) ^ 2]
Here, the arithmetic operations (addition, subtraction) - not applicable
For example: You have an average gap for the interval [0, x]. I mean a gap in the interval [m, x]. These two middle gap can not be compared. They are incompatible. You're trying hard to prove, based on its average gap that my average gap - is wrong.
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.
Why are your conclusions about the infinite growth of error does not show it back to the minimal error??
In this direction, so many possibilities, you are a priori declared all wrong.
E
0.5
0,3333333333333328
0.4
1,457142857142859
0,9740259740259744
1,016983016983016
1,997825703708054
1,732035766091498
4,039516950945141
2,953666772234264
4,363677766706288
2,40095579462409
4,375736627569869
5,018983639099712
3,22245558151018
13,4504870390029
0,107313851465896
11,05908622100744
5,552095561328978
6,805731948264758
8,954888824717338
5,644568909012201
12,88654617276976
17,89704622906506
6,291294056148678
6,603420310666865
12,09435208970917
8,486521705518053
2,825759803899164
31,62734124016704
18,51006382672811
16,69938859059135
12,91912926368558
21,73218035865914
8,37075320621024
21,06813232628563
23,63605686426048
15,24058793409204
22,04623796951981
31,49574008205725
0,783091165451072
38,52085515949986
4,023029872310733
19,72529352435859
7,700132644713038
48,16162388440734
• ... 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
Message 3 of 4 , Dec 10 9:42 AM
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<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
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