The Prime Problem

[Guest]

On The Theory Of Natural Numbers, we have the following general equations:

e = 2^a * ∏ [ from i = 1 to j of (pi)^ai ] (FTA works when pi > 1)

= p + q (GC)

= | s - r | (PC)

where e is any positive even integer, and a and ai are some integers ≥ 1;

pi, p, q, s, and r are odd primes ≥ 1.


FTA - Fundamental Theorem of Arithmetic;

GC - the sound Goldbach Conjecture; (see reference 2)

PC - the sound Polignac Conjecture; (see reference 2)

PNT - Prime Number Theorem;

PPL - Prime Parity Law (see reference 2)

SOP - the sound Sum of Primes Conjecture (see reference 2)

Note: All positive odd integers are encapsulated in all positive even integers.


The Prime Problem:

Discover an efficient algorithm which calculates primes and their corresponding indices in the natural sequence of natural numbers...

"There's the problem (above). Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus." --David Hilbert

"Simple seeks simplest (best) solution ..."


Keywords: The Scientific Method; N is the set of natural numbers;
Optimization theory and methods; algorithm R & D; AKS primality test; Number Theory;
Analysis and Synthesis.

Scientific Method Approach To Problem-Solving:

I. Hypothesis: Since prime numbers generate the positive even integers efficiently according to GC, PC, PNT, and PPL, the generation of prime numbers is the result of a process of integer optimization.

II. Test the hypothesis: We shall conduct an experiment to solve the prime problem via integer optimization using an improvised min-max principle.

III. The Experiment: Select some positive even integer, n ≥ 20 and solve the following problem.

Minimize |P| subject to: [ |P| ∈ 2N and |P| ≤ n/2 such that P = {2j -1 | j ∈ N and j ≤ n/2};

and Maximize |En| subject to: [
En = {m ≤ n | m∈ 2N; m = p + q for some p, q ∈ P such that
s/Log[s] ≥ k^2 + k (SOP Conjecture) where s = ∑( pi from i = 1 to i = |P| = 2k) where pi ∈ P } ]
].

IV. What are the results?

V. Questions:

1. Are all p ∈ P prime?

2. Is |P| minimum?

3. Is |En| maximum?

4. Is the hypothesis correct?

5. If the hypothesis is wrong, can we modify it to agree with the results? ...

References:
1. http://www.ieor.berkeley.edu/~hochbaum/ ... 9-2010.pdf
2. https://www.physforum.com/index.php?sho ... 0106&st=60

David Cole
aka primework123

[Guest]

It seems that the links do not work.

[Guest]

...

'refence 1 = 'A Minimax Strategy for Global Optimization' by STEFAN JAKOBSSON, et al. (google it)
refence 2 = 'On The Distribution of Prime Numbers' ... (google primework123 for more info.)
Please support my research efforts at http://gofundme.com/david_cole
Thank you!

[Guest]

It seems that the links do not work.

'refence 1 = 'A Minimax Strategy for Global Optimization' by STEFAN JAKOBSSON, et al. (google it)
refence 2 = 'On The Distribution of Prime Numbers' ... (google primework123 for more info.)
Please support my research efforts at http://gofundme.com/david_cole
Thank you!

[primework123]

On The Theory Of Natural Numbers, we have the following general equations:

e = 2^a * ∏ [ from i = 1 to j of (pi)^ai ] (FTA works when pi > 1)

= p + q (GC)

= | s - r | (PC)

where e is any positive even integer, and a and ai are some integers ≥ 1;

pi, p, q, s, and r are odd primes ≥ 1.


FTA - Fundamental Theorem of Arithmetic;

PCF - Prime-Counting Function - π(*)

GC - the sound Goldbach Conjecture;

PC - the sound Polignac Conjecture;

PNT - Prime Number Theorem;

PPL - Prime Parity Law [ π(e = m*g = 1 + p2n) = 2 * π(g = 1 + pn) = 2n where 2 < m ≤ 3 ];

SOP - the sound Sum of Primes Conjecture.

Note: All positive odd integers are encapsulated in all positive even integers.


The Prime Problem:

Discover an efficient algorithm which calculates primes and their corresponding indices in the natural sequence of natural numbers.

"There's the problem (above). Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus."
-- David Hilbert

"Simple seeks simplest (best) solution ..."

******Update********
Keywords: The Scientific Method; N is the set of natural numbers;
Optimization theory and methods; AKS primality test; Modern Methods of Number Theory; Complexity and Change;
Analysis and Synthesis; Algorithm R & D;
**********************

I. Hypothesis:
Since prime numbers generate the positive even integers efficiently according to GC, PC, PNT, and PPL, the generation of prime numbers is the result of a process of integer optimization.

II. Test the hypothesis:
We shall conduct an experiment to solve the prime problem via integer optimization using an improvised min-max principle.

III. The Experiment:
Select some positive even integer, n ≥ 20 and solve the following problem.

*****Update*********
Minimize |P| subject to: [ |P| ∈ 2N and |P| ≤ n/2 such that P = {2j -1 | j ∈ N and j ≤ n/2};
and
Maximize |En| subject to: [
En = {m ≤ n | m∈ 2N; m = p + q for some p, q ∈ P such that s/Log(s) = (k+[tex]\Delta[/tex]k)^2 + k+[tex]\Delta[/tex]k (SOP Conjecture)
with ∆k ≈ .00568272*k - 635.175 (tentatively) and where s = ∑( pi from i = 1 to i = |P| = 2k) where pi ∈ P } ]
].
**********************
IV. What are the results?

V. Questions:

1. Are all p ∈ P prime?

2. Is |P| minimum?

3. Is |En| maximum?

4. Is the hypothesis correct?

5. If the hypothesis is wrong, can we modify it to agree with the results? ...

P.S. Keep the faith (effort and hope) and keep an open mind. Thank Lord GOD!


David Cole
aka primework123
https://www.gofundme.com/david_cole
Thank you!

[primework123]

On The Theory Of Natural Numbers, we have the following general equations:

e = [tex]2^{a}[/tex] * ∏ ( from i = 1 to j of [tex]p_{i }^{a_{i } }[/tex] ) (FTA works when [tex]p_{i } > 1[/tex])

= p + q (GC)

= | s - r | (PC)

where e is any positive even integer; a and [tex]a_{i }[/tex] are some integers ≥ 1;

[tex]p_{i }[/tex], p, q, s, and r are odd primes ≥ 1.


FTA - Fundamental Theorem of Arithmetic;

PCF - Prime-Counting Function - π(*)

GC - the sound Goldbach Conjecture;

PC - the sound Polignac Conjecture;

PNT - Prime Number Theorem;

PPL - Prime Parity Law [ π(e = m*g = 1 + p2n) = 2 * π(g = 1 + pn) = 2n where 2 < m ≤ 3 ];

SOP - the sound Sum of Primes Conjecture.

Note: All positive odd integers are encapsulated in all positive even integers.


The Prime Problem:

Discover an efficient algorithm which calculates primes and their corresponding indices in the natural sequence of natural numbers.

"There's the problem (above). Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus."
-- David Hilbert

"Simple seeks simplest (best) solution ..."


Keywords: The Scientific Method; N is the set of natural numbers;
Optimization theory and methods; AKS primality test; Modern Methods of Number Theory; Complexity and Change;
Analysis and Synthesis; Algorithm R & D.

Scientific Method Approach To Problem-Solving:

I. Hypothesis:

Since prime numbers generate the positive even integers efficiently according to GC, PC, PNT, and PPL, the generation of prime numbers is the result of a process of integer optimization.

II. Test the hypothesis:
We shall conduct an experiment to solve the prime problem via integer optimization using an improvised min-max principle.

III. The Experiment:
Select some positive even integer, n ≥ 20 and solve the following problem.


Minimize |P| subject to: [ |P| ∈ 2N and |P| ≤ n/2 such that P = {2j -1 | j ∈ N and j ≤ n/2};
and
Maximize |En| subject to: [
En = {m ≤ n | m∈ 2N; m = p + q for some p, q ∈ P such that [tex]s/Log(s) = (k+\Delta k)^2 + k+\Delta k[/tex] (SOP Conjecture)
with ∆k ≈ .00568272*k - 635.175 for 100,000 ≤ k ≤ 2,000,000 and with s = ∑( [tex]p_{i }[/tex] from i = 1 to i = |P| = 2k ≤ n/2) where [tex]p_{i }[/tex] ∈ P } ] ].

IV. What are the results?

V. Questions:

1. Are all p ∈ P prime?

2. Is |P| minimum?

3. Is |En| maximum?

4. Is the hypothesis correct?

5. If the hypothesis is wrong, can we modify it to agree with the results? ...

P.S. Keep the faith (effort and hope) and keep an open mind. Thank Lord GOD!


David Cole
aka primework123
https://www.gofundme.com/david_cole
Thank you! Thank Lord GOD!

[Guest]

On The Theory Of Natural Numbers, we have the following general equations:

e = 2^a * ∏ [ from i = 1 to j of (p_i)^a_i ] (FTA works when p_i > 1)

= p + q (GC)

= | s - r | (PC)

where e is any positive even integer, and a and a_i are some integers ≥ 1;

p_i, p, q, s, and r are odd primes ≥ 1.


FTA - Fundamental Theorem of Arithmetic;

GC - the sound Goldbach Conjecture; (see reference 2)

PC - the sound Polignac Conjecture; (see reference 2)

PNT - Prime Number Theorem;

PPL - Prime Parity Law (see reference 2)

SOP - the sound Sum of Primes Conjecture (see reference 2)

Note: All positive odd integers are encapsulated in all positive even integers.


The Prime Problem:

Discover an efficient algorithm which calculates primes and their corresponding indices in the natural sequence of natural numbers...

"There's the problem (above). Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus." --David Hilbert

"Simple seeks simplest (best) solution ..."


Keywords: The Scientific Method; N is the set of natural numbers;
Optimization theory and methods; algorithm R & D; AKS primality test; Number Theory;
Analysis and Synthesis.

Scientific Method Approach To Problem-Solving:

I. Hypothesis: Since prime numbers generate the positive even integers efficiently according to GC, PC, PNT, and PPL, the generation of prime numbers is the result of a process of integer optimization.

II. Test the hypothesis: We shall conduct an experiment to solve the prime problem via integer optimization using an improvised min-max principle.

III. The Experiment: Select some positive even integer, n ≥ 20 and solve the following problem.

Minimize |P| subject to: [ |P| ∈ 2N and |P| ≤ n/2 such that P = {2j -1 | j ∈ N and j ≤ n/2};

and Maximize |E_n| subject to: [
E_n = {m ≤ n | m∈ 2N; m = p + q for some p, q ∈ P such that
s/Log[s] ≥ k^2 + k (SOP Conjecture) where s = ∑( p_i from i = 1 to i = |P| = 2k) where p_i ∈ P } ]
].

IV. What are the results?

V. Questions:

1. Are all p ∈ P prime?

2. Is |P| minimum?

3. Is |E_n| maximum?

4. Is the hypothesis correct?

5. If the hypothesis is wrong, can we modify it to agree with the results? ...

References:
1. http://www.ieor.berkeley.edu/~hochbaum/ ... 9-2010.pdf
2. https://www.physforum.com/index.php?sho ... 0106&st=60

David Cole
aka primework123