Maximum Prime Gap...

[Guest]

Estimate Of Maximum Prime Gap (MPG) Between Consecutive Primes For Primes Less Than N:

If we let the average prime gap, a(n), in the interval, [tex][0, n] =[/tex]{ [tex]x, 0 \le x\le n \ge 97[/tex] }, be

a(n) = n / [tex]\pi(n)[/tex],

then we estimate mpg:

[tex]\sum_{i=1}^{⌊a(n)⌋ - ⌊log( a(n) )⌋ }2 * i \le mpg < (1 + a(n) - ⌊a(n)⌋) * \sum_{i=1}^{⌊a(n)⌋ - ⌊log( a(n) )⌋ }2 * i[/tex]

where mpg is rounded to the nearest positive even integer;


Please verify this latest result. Thank you!

Keywords: [tex]\pi(n)[/tex] is prime-counting function and Uncertainty Principle For Calculating Primes (UCP) (see another post under number theory)

Reference Data Link: https://primes.utm.edu/notes/gaps.html

David Cole (aka primework123)

The term, [tex](1 + a(n) - ⌊a(n)⌋)[/tex], can probably be replaced by 1.25 so that we have:

[tex]\sum_{i=1}^{⌊a(n)⌋ - ⌊log (a(n))⌋ }2 * i \le mpg < 1.25 * \sum_{i=1}^{⌊a(n)⌋ - ⌊log (a(n))⌋ }2 * i[/tex]

where mpg is rounded to the nearest positive even integer;

[tex]mpg' = 1.125 * \sum_{i=1}^{⌊a(n)⌋ - ⌊log (a(n))⌋ }2 * i[/tex]

where mpg' is rounded to the nearest positive even integer.


Please verify these latest results. Thank you!

Reference Data Link: https://primes.utm.edu/notes/gaps.html

David Cole (aka primework123)[/quote]

[Guest]

Keywords: Prime Number Theorem (PNT), the sound Goldbach Conjecture (GC),
[tex]\pi(*)[/tex] is the prime counting function; , Fundamental Theorem of Arithmetic, Number Theory, and Analysis/Synthesis,...

Prime Work:

(1) There are infinitely many more positive integers (even or odd) than there are prime numbers, or prime numbers have a zero density relative to the positive integers,
and
(2) prime numbers generate the positive even integers so efficiently that gaps between two consecutive prime numbers increase without bound.

(3) Prime Parity Law (PPL):

[tex]\pi(e = mg = 1 + p_{2n }) = 2 * \pi(g = 1 + p_{n }) = 2n[/tex] where

[tex]p_{n } > 2, p_{2n }[/tex] are odd prime numbers; [tex]2 < m ≤ 3[/tex]; and as [tex]g\rightarrow\infty[/tex], [tex]m \rightarrow 2[/tex].

...

Note: mpg' ( the maximum prime gap between two consecutive prime numbers in the interval, [0, x ≥ 97] ) is calculated in accordance to the above laws for the distribution of prime numbers.

Let's Formulate an Uncertainty Principle (UCP) For Calculating the Exact Distribution of Odd Prime Numbers
(Hey, Quantum Physics has one!) until we can do better...

Keywords: Pi()--Odd Prime Counting Function, Li()--Logrithmatic Integral, ln()--natural log function, and pn and pn+1 are nth prime and (n+1)th prime, respectively, and we count 1 as prime only in the additive sense of number theory.

Here are some important facts we might considered provided Riemann Hypothesis (RH) is true (RH is true!):

*FACT I: |Pi(x) - Li(x)| < ( sqrt(x) * ln(x) ) / (8*[tex]\pi[/tex]) for all x ≥ 2657.

(*See Reference: Schoenfeld, Lowell (1976), "Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II",
Mathematics of Computation 30 (134): 337–360, doi:10.2307/2005976, JSTOR 2005976, MR 0457374.)

FACT II: (?) or could it be the Average Prime Gap Error (APGE) associated with the true average gap for all pn and pn+1 in the interval,
[1, x]?

FACT III: What could it be?

Uncertainty Principle (UCP) for the distribution of odd prime numbers:
UCP is |Pi(x) - Li(x)| * APGE ≥ ? for all x ≥ 2657.

Right? or Not even wrong!... !?

A Solution:

Uncertainty Principle (UCP) is |Pi(x) - Li(x)| * |a'(x) - a(x)|.

Calculation:

We let a'(x) = x/Pi(x) where a'(x) is the true average prime gap for all pn and pn+1 in [1, x].
And we let a(x) = x/Li(x) where a(x) is the approximate average prime gap for all pn and pn+1 in [1, x].
|Pi(x) - Li(x)| * |a'(x) - a(x) | < ? where
|Pi(x) - Li(x)| < sqrt(x) * ln(x)/(8*[tex]\pi[/tex]) for all x ≥ 2657, and APGE is |a'(x) - a(x)|, and [tex]\pi[/tex] = 3.14...

|Pi(x) - Li(x)| < sqrt(x) * ln(x)/(8*[tex]\pi[/tex]) implies Pi(x) = Li(x) + r * sqrt(x) * ln(x) / (8 *pi) where -1 < r < 1.

This last equation implies Pi(x) = (8 * [tex]\pi[/tex] * Li(x) + r * sqrt(x) * ln(x) )/(8 *[tex]\pi[/tex]).

Therefore, a'(x) = x / Pi(x) = 8 * pi * x / (8 * [tex]\pi[/tex] * Li(x) + r * sqrt(x) * ln(x) ).

So, |a'(x) - a(x)| = |8 * [tex]\pi[/tex] * x / (8 * [tex]\pi[/tex] * Li(x) + r * sqrt(x) * ln(x)) - x / Li(x) |
= x / Li(x) * | (8 * [tex]\pi[/tex] * Li(x) - 8 * [tex]\pi[/tex] * Li(x) - r * sqrt(x) * ln(x))/ (8 *[tex]\pi[/tex] * Li(x) + r * sqrt(x) * ln(x)) |
= x / Li(x) * | ( - r * sqrt(x) * ln(x))/ (8 * [tex]\pi[/tex] * Li(x) + r * sqrt(x) * ln(x)) |
< x / Li(x) * | ( sqrt(x) * ln(x))/ (8 * [tex]\pi[/tex] * Li(x) - sqrt(x) * ln(x)) |
= x / Li(x) * ( sqrt(x) * ln(x))/ (8 * [tex]\pi[/tex] * Li(x) - sqrt(x) * ln(x)) if r = -1.

Therefore, APGE is |a'(x) - a(x)| < x / Li(x) * | ( sqrt(x) * ln(x))/ (8 * [tex]\pi[/tex] * Li(x) - sqrt(x) * ln(x)) |.

Hence,
|Pi(x) - Li(x)| * |a'(x) - a(x) | < (sqrt(x) * ln(x)) /(8*[tex]\pi[/tex])) * x / Li(x) * ( sqrt(x) * ln(x))/ (8 *[tex]\pi[/tex] * Li(x) - sqrt(x) * ln(x))
= 1/(8 * [tex]\pi[/tex]) * 1/Li(x) * (x * ln(x))^2 / (8 * [tex]\pi[/tex] * Li(x) - sqrt(x) * ln(x)).

Thus, we have the following calculation of UCP**:

UCP is |Pi(x) - Li(x)| * |a'(x) - a(x) | < (x * ln(x) )^2 /[8 * [tex]\pi[/tex] * Li(x) * ( 8 * [tex]\pi[/tex] * Li(x) - sqrt(x) * ln(x) ) ].

**Note: The Riemann Prime-Counting Function R(x) will improve UCP substantially because R(x) = Pi(x) since it incorporates all the
non-trivial zeta zeros of the Riemann zeta function in its calculation.

David Cole
(aka primework123)
Please support my research work at: https://www.gofundme.com/david_cole
Thank you! Thank Lord GOD!
http://biblia.com/verseoftheday/image/Ro8.28[/quote]

[Guest]

Let assume [tex]mpg = 75,000,000[/tex], what are a(x), and x, approximately?

We have 75,000,000/1.25 = [tex]2 *\sum_{i=1}^{j }i =[/tex] 2*(1+j) * j/2 = j * (j+1).

This implies j [tex]\approx[/tex] 7482 = a(x) - ⌊log(a(x))⌋.

And therefore, [tex]a(x) \approx[/tex] 7491 = x /[tex]\pi(x) \approx x /Li(x)[/tex].

So, [tex]x \approx 7491 * Li(x).[/tex]

What is x, approxiately? Thank you!

Keyword: Li(*) is the Logarithmic Integral.

David Cole (aka primework123)

[Guest]

Let assume [tex]mpg = 75,000,000[/tex], what are a(x), and x, approximately?

We have 75,000,000/1.25 = [tex]2 *\sum_{i=1}^{j }i =[/tex] 2*(1+j) * j/2 = j * (j+1).

This implies j [tex]\approx[/tex] 7482 = a(x) - ⌊log(a(x))⌋.

And therefore, [tex]a(x) \approx[/tex] 7491 = x /[tex]\pi(x) \approx x /Li(x)[/tex].

So, [tex]x \approx 7491 * Li(x).[/tex]

What is x, approximately? Thank you!

Keyword: Li(*) is the Logarithmic Integral.

David Cole (aka primework123)

[Guest]

Let assume [tex]mpg = 75,000,000[/tex], what are a(x), and x, approximately?

We have 75,000,000/1.25 = [tex]2 *\sum_{i=1}^{j }i =[/tex] 2*(1+j) * j/2 = j * (j+1).

This implies j [tex]\approx[/tex] 7482 = a(x) - ⌊log(a(x))⌋.

And therefore, [tex]a(x) \approx[/tex] 7491 = x /[tex]\pi(x) \approx x /Li(x)[/tex].

So, [tex]x \approx 7491 * Li(x).[/tex]

What is x, approximately? Thank you!

Keyword: Li(*) is the Logarithmic Integral.

David Cole (aka primework123)

[Guest]

Let assume [tex]mpg = 75,000,000[/tex], what are a(x), and x, approximately?

We have 75,000,000/1.25 = [tex]2 *\sum_{i=1}^{j }i =[/tex] 2*(1+j) * j/2 = j * (j+1).

This implies j [tex]\approx[/tex] 7482 = a(x) - ⌊log(a(x))⌋.

And therefore, [tex]a(x) \approx[/tex] 7491 = x /[tex]\pi(x) \approx x /Li(x)[/tex].

So, [tex]x \approx 7491 * Li(x).[/tex]

What is x, approximately? Thank you!

Keyword: Li(*) is the Logarithmic Integral.

David Cole (aka primework123)

[tex]a(x) \approx[/tex] 7491 = x /[tex]\pi(x) \approx x /(x/log(x)) = log(x)[/tex].

So, [tex]x \approx e^{7491} \approx 1.995 * 10^{3253}.[/tex]

[Guest]

[tex]a(x) \approx[/tex] 7491 = x /[tex]\pi(x) \approx x /(x/log(x)) = log(x)[/tex].

So, [tex]x \approx e^{7491} \approx 1.995 * 10^{3253}.[/tex]

So, [tex]mpg/x \approx 0[/tex].

Therefore, the size of 'sufficiently large' prime gaps between consecutive primes is extremely small relative to the size of the their corresponding primes. This indicates there's much redundancy or repetition of previously generated prime gaps according to the Harmonic Series and PNT. In addition, I see the growth of new clusters or galaxies of primes separated by growing space between them like our expanding universe...

So, primes are relatively close together on average in their corresponding clusters while their corresponding clusters are growing farther and farther apart.

It's amazing how natural numbers mirror our natural world! Praise Lord GOD!

[Guest]

[tex]a(x) \approx[/tex] 7491 = x /[tex]\pi(x) \approx x /(x/log(x)) = log(x)[/tex].

So, [tex]x \approx e^{7491} \approx 1.995 * 10^{3253}.[/tex]

So, [tex]mpg/x \approx 0[/tex].

Therefore, the size of 'sufficiently large' prime gaps between consecutive primes is extremely small relative to the size of the their corresponding primes. This indicates there's much redundancy or repetition of previously generated prime gaps according to the Harmonic Series and PNT. In addition, I see the growth of new clusters or galaxies of primes separated by growing space between them like our expanding universe...

So, primes are relatively close together on average in their corresponding clusters while their corresponding clusters are growing farther and farther apart.

It's amazing how natural numbers mirror our natural world! Praise Lord GOD!

We have a(x) = 7491, [tex]x = 1.995 * 10^{3253}[/tex], when we have mpg/1.25 = 75,000,000/1.25;

We have a(x) = 8660, [tex]x = 9.777 * 10^{3760}[/tex], when we have mpg = 75,000,000.

So we expect to find a prime gap of 75,000,000 between consecutive primes in the range, [tex]1.995 * 10^{3253} \le x \le 9.777 * 10^{3760}[/tex].

There are roughly [tex]1.13*10^{3757}[/tex] primes in that range, and it will take a computer which
does [tex]10^{18}[/tex] calculations per second, at least [tex]3.58*10^{3731}[/tex] years to search all those primes!

[Guest]

[tex]a(x) \approx[/tex] 7491 = x /[tex]\pi(x) \approx x /(x/log(x)) = log(x)[/tex].

So, [tex]x \approx e^{7491} \approx 1.995 * 10^{3253}.[/tex]

So, [tex]mpg/x \approx 0[/tex].

Therefore, the size of 'sufficiently large' prime gaps between consecutive primes is extremely small relative to the size of the their corresponding primes. This indicates there's much redundancy or repetition of previously generated prime gaps according to the Harmonic Series and PNT. In addition, I see the growth of new clusters or galaxies of primes separated by growing space between them like our expanding universe...

So, primes are relatively close together on average in their corresponding clusters while their corresponding clusters are growing farther and farther apart.

It's amazing how natural numbers mirror our natural world! Praise Lord GOD!

We have a(x) = 7491, [tex]x = 1.995 * 10^{3253}[/tex], when we have mpg/1.25 = 75,000,000/1.25;

We have a(x) = 8660, [tex]x = 9.777 * 10^{3760}[/tex], when we have mpg = 75,000,000.

So we expect to find a prime gap of 75,000,000 between consecutive primes in the range, [tex]1.995 * 10^{3253} \le x \le 9.777 * 10^{3760}[/tex].

There are roughly [tex]1.13*10^{3757}[/tex] primes in that range, and it will take a computer which
does [tex]10^{18}[/tex] calculations per second, at least [tex]3.58*10^{3731}[/tex] years to search all those primes!

Correction: We have a(x) = 8651 instead of 8660, ..., and it take the above computer at least [tex]4.42*10^{3727}[/tex] years to search all those primes!

[Guest]

[tex]a(x) \approx[/tex] 7491 = x /[tex]\pi(x) \approx x /(x/log(x)) = log(x)[/tex].

So, [tex]x \approx e^{7491} \approx 1.995 * 10^{3253}.[/tex]

So, [tex]mpg/x \approx 0[/tex].

Therefore, the size of 'sufficiently large' prime gaps between consecutive primes is extremely small relative to the size of the their corresponding primes. This indicates there's much redundancy or repetition of previously generated prime gaps according to the Harmonic Series and PNT. In addition, I see the growth of new clusters or galaxies of primes separated by growing space between them like our expanding universe...

So, primes are relatively close together on average in their corresponding clusters while their corresponding clusters are growing farther and farther apart.

It's amazing how natural numbers mirror our natural world! Praise Lord GOD!

We have a(x) = 7491, [tex]x = 1.995 * 10^{3253}[/tex], when we have mpg/1.25 = 75,000,000/1.25;

We have a(x) = 8660, [tex]x = 9.777 * 10^{3760}[/tex], when we have mpg = 75,000,000.

So we expect to find a prime gap of 75,000,000 between consecutive primes in the range, [tex]1.995 * 10^{3253} \le x \le 9.777 * 10^{3760}[/tex].

There are roughly [tex]1.13*10^{3757}[/tex] primes in that range, and it will take a computer which
does [tex]10^{18}[/tex] calculations per second, at least [tex]3.58*10^{3731}[/tex] years to search all those primes!

Correction: We have a(x) = 8651 instead of 8660, ..., and it take the above computer at least [tex]4.42*10^{3727}[/tex] years to search all those primes!

mpg = 75,000,000 is roughly [tex]log(x = 9.777 * 10^{3760})^{2}[/tex]

Could it be that for 'sufficiently large' prime gaps (gp) for consecutive primes of size x, we mean [tex]gp >> log(x)^{2}[/tex]?

But this property, 'sufficiently large' prime gap, is dependent on the size of consecutive primes according to theory (see above mpg conjecture, PNT, RH, Harmonic Series, the sound Polignac's Conjecture, and the distribution of prime numbers, etc.)

[Guest]

Do you see some parallels between physics (quantum and cosmology) and prime number theory on the topics discussed previously?

I hope so.

What are they? Please do the homework. There will be a test later!

P.S. From 3D to 2D to 1D; Good things are happening in the D!

[Guest]

[tex]a(x) \approx[/tex] 7491 = x /[tex]\pi(x) \approx x /(x/log(x)) = log(x)[/tex].

So, [tex]x \approx e^{7491} \approx 1.995 * 10^{3253}.[/tex]

So, [tex]mpg/x \approx 0[/tex].

Therefore, the size of 'sufficiently large' prime gaps between consecutive primes is extremely small relative to the size of the their corresponding primes. This indicates there's much redundancy or repetition of previously generated prime gaps according to the Harmonic Series and PNT. In addition, I see the growth of new clusters or galaxies of primes separated by growing space between them like our expanding universe...

So, primes are relatively close together on average in their corresponding clusters while their corresponding clusters are growing farther and farther apart.

It's amazing how natural numbers mirror our natural world! Praise Lord GOD!

We have a(x) = 7491, [tex]x = 1.995 * 10^{3253}[/tex], when we have mpg/1.25 = 75,000,000/1.25;

We have a(x) = 8660, [tex]x = 9.777 * 10^{3760}[/tex], when we have mpg = 75,000,000.

So we expect to find a prime gap of 75,000,000 between consecutive primes in the range, [tex]1.995 * 10^{3253} \le x \le 9.777 * 10^{3760}[/tex].

There are roughly [tex]1.13*10^{3757}[/tex] primes in that range, and it will take a computer which
does [tex]10^{18}[/tex] calculations per second, at least [tex]3.58*10^{3731}[/tex] years to search all those primes!

Correction: We have a(x) = 8651 instead of 8660, ..., and it take the above computer at least [tex]4.42*10^{3727}[/tex] years to search all those primes!

mpg = 75,000,000 is roughly [tex]log(x = 9.777 * 10^{3760})^{2}[/tex]

Could it be that for 'sufficiently large' prime gaps (gp) for consecutive primes of size x, we mean [tex]gp >> log(x)^{2}[/tex]?

But this property, 'sufficiently large' prime gap, is dependent on the size of consecutive primes according to theory (see above mpg conjecture, PNT, RH, Harmonic Series, the sound Polignac's Conjecture, and the distribution of prime numbers, etc.)[/quote]

There's an excellent paper on maximum prime gaps with excellent references at the following link:

http://arxiv.org/ftp/arxiv/papers/1312/1312.2481.pdf
('Note On The Maximal Prime Gaps' by N. A. Carella)

[Guest]

[tex]a(x) \approx[/tex] 7491 = x /[tex]\pi(x) \approx x /(x/log(x)) = log(x)[/tex].

So, [tex]x \approx e^{7491} \approx 1.995 * 10^{3253}.[/tex]

So, [tex]mpg/x \approx 0[/tex].

Therefore, the size of 'sufficiently large' prime gaps between consecutive primes is extremely small relative to the size of the their corresponding primes. This indicates there's much redundancy or repetition of previously generated prime gaps according to the Harmonic Series and PNT. In addition, I see the growth of new *clusters or galaxies of primes separated by growing space between them like our expanding universe...

So, primes are relatively close together on average in their corresponding clusters while their corresponding clusters are growing farther and farther apart.

It's amazing how natural numbers mirror our natural world! Praise Lord GOD!

We have a(x) = 7491, [tex]x = 1.995 * 10^{3253}[/tex], when we have mpg/1.25 = 75,000,000/1.25;

We have a(x) = 8660, [tex]x = 9.777 * 10^{3760}[/tex], when we have mpg = 75,000,000.

So we expect to find a prime gap of 75,000,000 between consecutive primes in the range, [tex]1.995 * 10^{3253} \le x \le 9.777 * 10^{3760}[/tex].

There are roughly [tex]1.13*10^{3757}[/tex] primes in that range, and it will take a computer which
does [tex]10^{18}[/tex] calculations per second, at least [tex]3.58*10^{3731}[/tex] years to search all those primes!

Correction: We have a(x) = 8651 instead of 8660, ..., and it take the above computer at least [tex]4.42*10^{3727}[/tex] years to search all those primes!

*Note: Reference Link:

http://complex.ffn.ub.es/~mbogunya/arch%20...%20022806.pdf
(Complex architecture of primes and natural numbers)

[Guest]

Therefore, the size of 'sufficiently large' prime gaps between consecutive primes is extremely small relative to the size of the their corresponding primes. This indicates there's much redundancy or repetition of previously generated prime gaps according to the Harmonic Series and PNT. In addition, I see the growth of new *clusters or galaxies of primes separated by growing space between them like our expanding universe...

So, primes are relatively close together on average in their corresponding clusters while their corresponding clusters are growing farther and farther apart.

It's amazing how natural numbers mirror our natural world! Praise Lord GOD![/quote]
[b][size=150]

*Note: Reference Link:

http://complex.ffn.ub.es/~mbogunya/archivos_cms/files/PhysRevE.90.022806.pdf
(Complex architecture of primes and natural numbers)

[Guest]

Therefore, the size of 'sufficiently large' prime gaps between consecutive primes is extremely small relative to the size of the their corresponding primes. This indicates there's much redundancy or repetition of previously generated prime gaps according to the Harmonic Series and PNT. In addition, I see the growth of new *clusters or galaxies of primes separated by growing space between them like our expanding universe...

So, primes are relatively close together on average in their corresponding clusters while their corresponding clusters are growing farther and farther apart.

It's amazing how natural numbers mirror our natural world! Praise Lord GOD!

*Note: Reference Link:

http://complex.ffn.ub.es/~mbogunya/archivos_cms/files/PhysRevE.90.022806.pdf
(Complex architecture of primes and natural numbers)[/quote]

[quote="primework123"] (From Proof of Polignac's Conjecture...)

So, given [tex]π[\sqrt{m_{k } * r_{k }}][/tex]= [tex]π[\sqrt{m_{k+1 } * r_{k+1 }}][/tex] = [tex]π[\sqrt{m_{k+2 } * r_{k+2 }}][/tex] = ...

= [tex]π[\sqrt{m_{k+n } * r_{k+n }}][/tex] < [tex]π[\sqrt{m_{l } * r_{l }}][/tex] = [tex]π[\sqrt{m_{k } * r_{k }}][/tex] + 1

for many arbitrary long intervals (empty of primes) between **clusters of primes,

we shall find that [tex]∏ (π[\sqrt{m_{k } * r_{k }}] − 1 ) / π[\sqrt{m_{k } * r_{k }}] = 0[/tex] for k = 1 to k = [tex]\infty[/tex].

**Note: [tex]m_{k+1 } * r_{k+1 }[/tex], [tex]m_{k+2 } * r_{k+2 }[/tex], ..., [tex]m_{k+n } * r_{k+n }[/tex] is part of one of many such prime clusters.

Note: [tex]\sqrt{m_{1 } * r_{1 }}[/tex] < [tex]\sqrt{m_{2 } * r_{2 }}[/tex] < ... <[tex]\sqrt{m_{n } * r_{n}}[/tex]

since we have the exceptional primes, [tex]e_{1 }[/tex] < [tex]e_{2 }[/tex] < [tex]e_{2 }[/tex] < ... < [tex]e_{n }[/tex] for all integral n

where [tex]e_{n }[/tex] = [tex]m_{n } * r_{n}[/tex] + 2i.

Recall 2i is the prime gap between two consecutive odd primes that does not exist or repeats finitely many times in the natural sequence of the natural numbers.

[Guest]

Asymptotically, as [tex]x \rightarrow \infty, mpg \rightarrow log(x)^{2} = a(x)^{2}.[/tex] Right?

Recall mpg is the maximal prime gap between two consecutive prime numbers which are less than or equal to x.

Note: a(x) = x / [tex]\pi(x)[/tex]

[Guest]

Asymptotically, as [tex]x \rightarrow \infty, mpg \rightarrow log(x)^{2} = a(x)^{2}.[/tex] Right?

Recall mpg is the maximal prime gap between two consecutive prime numbers which are less than or equal to x.

Note: a(x) = x / [tex]\pi(x).[/tex]

Excellent Reference Link: (Cramér's conjecture)

https://en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture

[Guest]

[tex]a(x) \approx[/tex] 7491 = x /[tex]\pi(x) \approx x /(x/log(x)) = log(x)[/tex].

So, [tex]x \approx e^{7491} \approx 1.995 * 10^{3253}.[/tex]

So, [tex]mpg/x \approx 0[/tex].

Therefore, the size of 'sufficiently large' prime gaps between consecutive primes is extremely small relative to the size of the their corresponding primes. This indicates there's much redundancy or repetition of previously generated prime gaps according to the Harmonic Series and PNT. In addition, I see the growth of new *clusters or galaxies of primes separated by growing space between them like our expanding universe...

So, primes are relatively close together on average in their corresponding clusters while their corresponding clusters are growing farther and farther apart.

It's amazing how natural numbers mirror our natural world! Praise Lord GOD!

We have a(x) = 7491, [tex]x = 1.995 * 10^{3253}[/tex], when we have mpg/1.25 = 75,000,000/1.25;

We have a(x) = 8660, [tex]x = 9.777 * 10^{3760}[/tex], when we have mpg = 75,000,000.

So we expect to find a prime gap of 75,000,000 between consecutive primes in the range, [tex]1.995 * 10^{3253} \le x \le 9.777 * 10^{3760}[/tex].

There are roughly [tex]1.13*10^{3757}[/tex] primes in that range, and it will take a computer which
does [tex]10^{18}[/tex] calculations per second, at least [tex]3.58*10^{3731}[/tex] years to search all those primes!

Correction: We have a(x) = 8651 instead of 8660, ..., and it take the above computer at least [tex]4.42*10^{3727}[/tex] years to search all those primes!

*Note: Reference Link:

http://complex.ffn.ub.es/~mbogunya/arch%20...%20022806.pdf
(Complex architecture of primes and natural numbers)[/quote]

The Probabilistic Way:

Let [tex]p_{1 } =[/tex][tex]NextPrime(1.995 * 10^{3253})[/tex].

Is [tex]p_{1 } + 75,000,000[/tex] prime?

If yes, check to see if [tex]p_{1 } + 2i[/tex] is prime for some random positive integer i < 37,500, 000?

If [tex]p_{1 } + 2i[/tex] is not prime after a number of trials, then we have probably found a prime gap

of 75,000,000 between two consecutive primes, [tex]p_{1 }[/tex] and [tex]NextPrime( p_{1 } )[/tex].

(Step 1): If no, let [tex]p_{j } =[/tex][tex]NextPrime(1.995 * 10^{3253} + 75,000,000 * j)[/tex] for some random positive integer j.

Is [tex]p_{j } + 75,000,000[/tex] prime?

If yes, check to see if [tex]p_{j } + 2i[/tex] is prime for some random positive integer i < 37,500, 000?

If [tex]p_{j } + 2i[/tex] is not prime after a number of trials, then we have probably found a prime gap

of 75,000,000 between two consecutive primes, [tex]p_{j }[/tex] and [tex]NextPrime( p_{j } )[/tex].

If no, goto step 1 (choose randomly a different j value).

And we can devise an algorithm which can do the calculations in parallel involving several processors or supercomputers for a number of potential prime candidates:

[tex]p_{j } =[/tex][tex]NextPrime(1.995 * 10^{3253} + 75,000,000 * j).[/tex]

An we expect success in a very reasonable time (a few days or weeks of massively parallel computing with super processors).

[Guest]

Therefore, the size of 'sufficiently large' prime gaps between consecutive primes is extremely small relative to the size of the their corresponding primes. This indicates there's much redundancy or repetition of previously generated prime gaps according to the Harmonic Series and PNT. In addition, I see the growth of new *clusters or galaxies of primes separated by growing space between them like our expanding universe...

So, primes are relatively close together on average in their corresponding clusters while their corresponding clusters are growing farther and farther apart.

It's amazing how natural numbers mirror our natural world! Praise Lord GOD!

*Note: Reference Link:

http://complex.ffn.ub.es/~mbogunya/archivos_cms/files/PhysRevE.90.022806.pdf
(Complex architecture of primes and natural numbers)

(From Proof of Polignac's Conjecture...)

So, given [tex]π[\sqrt{m_{k } * r_{k }}][/tex]= [tex]π[\sqrt{m_{k+1 } * r_{k+1 }}][/tex] = [tex]π[\sqrt{m_{k+2 } * r_{k+2 }}][/tex] = ...

= [tex]π[\sqrt{m_{k+n } * r_{k+n }}][/tex] < [tex]π[\sqrt{m_{l } * r_{l }}][/tex] = [tex]π[\sqrt{m_{k } * r_{k }}][/tex] + 1

for many arbitrary long intervals (empty of primes) between **clusters of primes,

we shall find that [tex]∏ (π[\sqrt{m_{k } * r_{k }}] − 1 ) / π[\sqrt{m_{k } * r_{k }}] = 0[/tex] for k = 1 to k = [tex]\infty[/tex].

**Note (Correction): The sequence, [tex]m_{k+1 } * r_{k+1 }[/tex], [tex]m_{k+2 } * r_{k+2 }[/tex], ..., [tex]m_{k+n } * r_{k+n }[/tex], is part of one of many such intervals between prime clusters.

Note: [tex]\sqrt{m_{1 } * r_{1 }}[/tex] < [tex]\sqrt{m_{2 } * r_{2 }}[/tex] < ... <[tex]\sqrt{m_{n } * r_{n}}[/tex]

since we have the exceptional primes, [tex]e_{1 }[/tex] < [tex]e_{2 }[/tex] < [tex]e_{2 }[/tex] < ... < [tex]e_{n }[/tex] for all integral n

where [tex]e_{n }[/tex] = [tex]m_{n } * r_{n}[/tex] + 2i.

Recall 2i is the prime gap between two consecutive odd primes that does not exist or repeats finitely many times in the natural sequence of the natural numbers.

[Guest]

[tex]a(x) \approx[/tex] 7491 = x /[tex]\pi(x) \approx x /(x/log(x)) = log(x)[/tex].

So, [tex]x \approx e^{7491} \approx 1.995 * 10^{3253}.[/tex]

So, [tex]mpg/x \approx 0[/tex].

Therefore, the size of 'sufficiently large' prime gaps between consecutive primes is extremely small relative to the size of the their corresponding primes. This indicates there's much redundancy or repetition of previously generated prime gaps according to the Harmonic Series and PNT. In addition, I see the growth of new clusters or galaxies of primes separated by growing space between them like our expanding universe...

So, primes are relatively close together on average in their corresponding clusters while their corresponding clusters are growing farther and farther apart.

It's amazing how natural numbers mirror our natural world! Praise Lord GOD!