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