The conjecture proposes that within sets A and B, which are defined by arithmetic progressions, the ratio of primes to composites follows an obvious logarithmic relationship. Specifically, it suggests that as the density of composite numbers increases, the density of prime numbers decreases.

This relationship is characterized by a logarithmic function, ℎ(𝑛) = log(𝑃(𝑛)/𝑆(𝑛)), where 𝑃(𝑛) represents the number of prime numbers less than or equal to 𝑛 that are elements of A or B but not elements of the product sets (AA, AB, or BB), and 𝑆(𝑛) represents the number of composite numbers less than or equal to 𝑛 that are elements of the product sets.

In essence, the conjecture suggests that the distribution of primes within these sets can be described logarithmically, reflecting a balance between the increasing density of composite numbers and the decreasing density of primes.

Conjecture 1:

Premise:

Let A = {6x + 5 | x ∈ Z} and B = {6y + 7 | y ∈ Z} be sets of integers for all values of x and y including 0.
Define AA, AB, and BB as the product sets:
- AA = {(6x + 5)(6y + 5) | x, y ∈ Z}
- AB = {(6x + 5)(6y + 7) | x, y ∈ Z}
- BB = {(6x + 7)(6y + 7) | x, y ∈ Z}
Let P(n) be the number of prime numbers less than or equal to n that are elements of A or B but not elements of AA, AB, or BB.
Let S(n) be the number of composite numbers less than or equal to n that are elements of AA, AB, or BB.
We aim to show that h(n) = log(P(n)/S(n)) exists and reflects the relative frequency of prime numbers to composite numbers in Sets A and B below the limit of integer n.

Argument:

Primes in A and B: All primes greater than 3 are of the form 6k ± 1, guaranteeing that all primes (except 2 and 3) are contained within sets A and B.
Composite Dominance: As n increases, the composite numbers generated by AA, AB, and BB become increasingly dominant. This is because:
- Growth Rates: AA, AB, and BB are quadratic functions (due to the xy terms), while A and B are linear functions. Therefore, as n grows, AA, AB, and BB generate numbers at a faster rate.
- Density: The density of primes within sets A and B becomes diluted by the rapid increase in composite numbers generated by AA, AB, and BB. This is because the number of composite numbers in AA, AB, and BB grows much faster than the number of primes in A and B.
Logarithmic Relationship:
- PNT and Dirichlet’s Theorem: The Prime Number Theorem (PNT) and its extension, Dirichlet’s Theorem on Arithmetic Progressions, suggest a general logarithmic relationship between prime numbers and their distribution.
- Dominance of Composites: Because composite numbers generated by AA, AB, and BB dominate as n increases, the density of primes within A and B is effectively governed by this rapid growth of composites. This creates a logarithmic relationship between the number of primes and composites within A and B, as described by the function h(n).
Formal Proof: A formal proof would involve rigorously demonstrating the following:
- Density Calculation: Compute the density of primes within sets A, B, AA, AB, and BB for increasing values of n.
- Ratio Analysis: Analyze the ratio of the density of primes within A and B to the density of primes within AA, AB, and BB as n increases.
- Limit Behavior: Show that as n goes to infinity, the ratio of densities approaches a value that is related to the logarithmic function h(n).

Conclusion:

By combining the argument that composite numbers generated by AA, AB, and BB dominate as n increases with the broader context provided by the Prime Number Theorem and Dirichlet’s Theorem, we can infer that a logarithmic relationship, as described by h(n), exists between the number of primes and composites within sets A and B.

(Assuming Conjecture 1) Conjecture 2:

Let:

k be a positive integer.

Define:

P(n) as the number of prime numbers less than or equal to n, plus 1, 2, and 3.
S(n) as the number of composite numbers (after eliminating all redundancies) less than or equal to n, plus all non-prime multiples of 2k, 3k, and 5k (excluding 2 and 3) less than or equal to n.

Then:

A logarithmic function, ℎ(𝑛) = log(𝑃(𝑛)/𝑆(𝑛)), exists and precisely describes the relationship between the number of all prime numbers and all non-prime numbers as n approaches infinity.

Refined Conjecture 1:

As n approaches infinity, the influence of all numbers less than n on the existence of prime numbers in the 6k ± 1 forms becomes negligible. This implies that the existence of a prime number within any future iteration of the 6k ± 1 forms is always possible, regardless of the density of composites generated before n.

Formalized Statements:

lim_(n→∞) ρ(n - 1) / ρ(n) = 1 
lim_(n→∞) σ(n) / ρ(n) = ∞

ρ(n) is the density of primes in the sets A = {6k – 1} and B = {6k + 1} up to n.
σ(n) is the density of composite numbers generated by products from sets A and B up to n.

Proof:

1. Prime Density in 6k ± 1 Forms (ρ(n))

Prime Number Theorem (PNT): The PNT states that the number of primes less than n, denoted as π(n), is asymptotically equal to n / log(n).
Dirichlet’s Theorem: Dirichlet’s Theorem on arithmetic progressions guarantees that there are infinitely many primes in any arithmetic progression of the form a + nd where a and d are coprime.
Approximation of ρ(n): Since every prime p > 3 can be written as 6k ± 1, we can approximate the density of primes in these forms as:
```
ρ(n) ≈ 2π(n) / (3n) ≈ 2n / (3n log(n)) ≈ 2 / (3log(n))
```
2. Composite Density (σ(n))

Composite numbers in the forms 6k ± 1 are generated by the product of two primes in these forms.
As n increases, the number of composite numbers grows more rapidly due to the quadratic nature of the product.

3. Ratio ρ(n – 1) / ρ(n)

Using the PNT-based approximation for ρ(n):
```
ρ(n - 1) / ρ(n) ≈ (2 / (3log(n - 1))) / (2 / (3log(n)))  
                 ≈ log(n) / log(n - 1)
                 → 1 as n → ∞
```
This shows that as n approaches infinity, the influence of primes in the interval [1, n – 1] becomes negligible compared to the density of primes in the interval [1, n].

4. Ratio σ(n) / ρ(n)

Using approximations for σ(n) (which grows quadratically) and ρ(n) (which grows logarithmically):

σ(n) / ρ(n) ≈ (n^2 / log(n)) / (2n / (3log(n)))
             ≈ (3/2) * n 
             → ∞ as n → ∞

The conjecture is supported by the fact that while the density of primes decreases and composites increase, the mathematical properties of primes ensure that primes in 6k ± 1 forms always exist.

Refined Conjecture 2:

The conjecture proposes a relationship between the number of prime numbers and composite numbers (including specific non-prime multiples) described by a logarithmic function
ℎ(𝑛) = log⁡(𝑃(𝑛)/𝑆(𝑛)). Let’s define and analyze the terms and the proposed function more rigorously.

Definitions:

P(n): The number of prime numbers less than or equal to n, plus 1, 2, and 3.
- 𝑃(𝑛) = π(𝑛) + 3
- where π(𝑛) is the prime-counting function.
S(n): The number of composite numbers (after eliminating redundancies) less than or equal to n, plus all non-prime multiples of 2k, 3k, and 5k (excluding 2 and 3) less than or equal to n.
- To avoid redundancy, each composite number is counted once.
- Includes non-prime multiples of 2k, 3k, and 5k less than n.

Analysis:

Prime-Counting Function (π(𝑛))
- According to the Prime Number Theorem (PNT), π(𝑛) ~ n / log(n).
Composite Counting Function (C(n))
- C(n) can be approximated as n – π(n) since composites and primes partition the set of natural numbers.
- Additionally, we need to consider the non-prime multiples of 2k, 3k, and 5k less than or equal to n. This involves using the inclusion-exclusion principle to avoid overcounting.
Non-Prime Multiples of 2k, 3k, and 5k
- The number of multiples of 2k up to n is ⌊n / 2k⌋.
- Similarly, for 3k and 5k, it is ⌊n / 3k⌋ and ⌊n / 5k⌋, respectively.
Function h(n)
- The function h(n) = log⁡(𝑃(𝑛)/𝑆(𝑛)) aims to describe the relationship between primes and composites as n → ∞.

Asymptotic Behavior:

As n → ∞:

P(n) ~ n / log(n) + 3
S(n) includes n – π(n) plus the additional non-prime multiples of 2k, 3k, and 5k. The dominant term is n – π(n), which simplifies to n asymptotically because π(n) grows slower than n.

Therefore, asymptotically:

P(n) / S(n) ~ (n / log(n) + 3) / (n + ⌊n / 2k⌋ + ⌊n / 3k⌋ + ⌊n / 5k⌋ - π(n)) 
           ~ 1 / log(n)

since the additional terms become negligible as n grows.

Logarithmic Function:
- h(n) = log⁡(𝑃(𝑛)/𝑆(𝑛)) ~ log⁡(1 / log(n)) = -log⁡(log(n))

Conclusion:

The refined conjecture can be stated as follows:

Conjecture 2 (Refined):

Let k be a positive integer. Define:

P(n) as the number of prime numbers less than or equal to n, plus 1, 2, and 3.
S(n) as the number of composite numbers (after eliminating redundancies) less than or equal to n, plus all non-prime multiples of 2k, 3k, and 5k (excluding 2 and 3) less than or equal to n.

Then, the logarithmic function h(n) = log⁡(𝑃(𝑛)/𝑆(𝑛)) describes the relationship between the number of all prime numbers and all non-prime numbers as n approaches infinity, and asymptotically:

h(n) ~ -log(log(n))

This refined conjecture captures the asymptotic behavior of the ratio of primes to composites, with h(n) approaching -log(log(n)) as n grows large.

See: 1,2,3,4,5