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 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 + p2n) = 2 * π(g = 1 + pn) = 2n

where π(*) is the prime-counting function;  and pn > 2 and p2n 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.

*****

https://en.wikipedia.org/wiki/Goldbach%27s_conjecture

https://en.wikipedia.org/wiki/Polignac%27s_conjecture

https://www.researchgate.net/post/What_is_the_correct_proof_of_the_famous_and_important_Polignac_Conjecture

https://www.researchgate.net/post/Why_is_the_stronger_Cramer_Conjecture_by_Daniel_Shanks_and_Andrew_Granville_true