IMPLICATIONS OF THE PRIME NUMBER THEOREM (PNT)

[Guest]

Keywords: Prime Number Theorem (PNT), the sound Riemann Hypothesis (RH) and the Riemann Prime-Counting Function, the sound Goldbach Conjecture (GC), the sound Polignac Conjecture (PC), the sound Prime Parity Law (PPL), and the purpose of prime numbers -- PRIME WORK (PW).

Prime Number Theorem (PNT) is true if and if only the Goldbach Conjecture (GC) and the Polignac Conjecture (PC) are true.

PRIME WORK (PW):

(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 according to the Prime Number Theorem (PNT).

(2) Prime numbers generate the positive even integers so efficiently according to the Prime Number Theorem (PNT) that gaps between two consecutive prime numbers increase in size without bound if and only if the Goldbach Conjecture (GC) and the Polignac Conjecture (PC) are true.

(3) Prime Parity Law: (PPL)

π( e = m*g = 1 + [tex]p_{2n }[/tex] ) = 2 * π( g = 1 + [tex]p_{n }[/tex] ) = 2n

where π(*) is the prime-counting function;

[tex]p_{n }[/tex] > 2 and [tex]p_{2n }[/tex] are odd prime numbers;

2 < m ≤ 3

where one is unit prime; and as g → ∞, m → 2.

*****

"Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty, spiritual formulas are discovered necessarily for the deeper penetration into the laws of nature." -- Albert Einstein.

*****

Reference links:
Goldbach Conjecture, https://en.wikipedia.org/wiki/Goldbach%27s_conjecture;
Polignac Conjecture, https://en.wikipedia.org/wiki/Polignac%27s_conjecture.

David Cole
aka primework123

*****

[Guest]

My research webpage link is:

https://www.researchgate.net/profile/David_Cole24.

It posts some interesting questions with some answers, and there are also several proofs of important conjectures too and more ... Please enjoy.

[Guest]

This may be last post here. I don't feel well (possibly food poisoning).

But I want to thank the staff of Math10.com for allowing to post my work here with much freedom. I hope my work is a significant contribution to mathematical science (prime number theory). Please pray for me and pray for world peace too.

Best wishes and peace,

David Cole

aka primework123

[Guest]

I discover a correction below:
Keywords: Prime Number Theorem (PNT), the sound Riemann Hypothesis (RH) and the Riemann Prime-Counting Function, the sound Goldbach Conjecture (GC), the sound Polignac Conjecture (PC), the sound Prime Parity Law (PPL), and the purpose of prime numbers -- PRIME WORK (PW).

(I discovered a mistake or typo in the statement below, and I corrected it.)
Prime Number Theorem (PNT) is true if and only if the Goldbach Conjecture (GC) and the Polignac Conjecture (PC) are true.

PRIME WORK (PW):

(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 according to the Prime Number Theorem (PNT).

(2) Prime numbers generate the positive even integers so efficiently according to the Prime Number Theorem (PNT) that gaps between two consecutive prime numbers increase in size without bound if and only if the Goldbach Conjecture (GC) and the Polignac Conjecture (PC) are true.

(3) Prime Parity Law: (PPL)

π( e = m*g = 1 + [tex]p_{2n }[/tex] ) = 2 * π( g = 1 + [tex]p_{n }[/tex] ) = 2n

where π(*) is the prime-counting function;

[tex]p_{n }[/tex] > 2 and [tex]p_{2n }[/tex] are odd prime numbers;

2 < m ≤ 3

where one is unit prime; and as g → ∞, m → 2.

*****

"Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty, spiritual formulas are discovered necessarily for the deeper penetration into the laws of nature." -- Albert Einstein.

*****

Reference links:
Goldbach Conjecture, https://en.wikipedia.org/wiki/Goldbach%27s_conjecture;
Polignac Conjecture, https://en.wikipedia.org/wiki/Polignac%27s_conjecture.

David Cole
aka primework123

*****

[leesajohnson]

Thank you to all for sharing such a useful theorem.

[Guest]

Lets see the Secrets of Prime Numbers
when I divide 1 to prime numbers I see the same occurence for example 1/13.
Our number is 0.076923.
76923 x 1 = 076 923
76 923 x 2 = 153 846
76 923 x 3 = 230 769
76 923 x 4 = 307 692
76 923 x 5 = 384 615
76 923 x 6 = 461 538
76 923 x 7 = 538 461
76 923 x 8 = 615384
76,923 x 9 = 692 307
76923 x 10 = 769230
76923 x 11 = 846153
76923 x 12 = 923076
76923 x 13 = 999999
can you see the iteration of same numbers 076923
and did you notice the 076+923=999
I did it for other prime numbers and saw more strange results

https://wordpress.com/block-editor/page ... ss.com/143