Overview
The main takeaway from the following is that a twin prime must be of the form (p,p+2) and (A,B) simultaneously.
Due to the modulo classes from which Set A and Set B arise, there can never be an integer which is a member of both Set A and Set B (they are “disjoint”).
Despite this disjointness, the Set A and Set B have equal cardinality, so that the absolute value of A is equal to the absolute value of B in the symmetric range -q<0<q, eg |A|=|B|.
Due to the different modulo classes from which composites of Set A and Set B emerge (that is AA, AB, and BB); that AB numbers can only be in Set A (because they are also of the form ≡ 5 (mod 6)); and AA and BB numbers can only be in Set B (because they are also of the form ≡ 1 (mod 6)).
However, when a negative number is considered in Set A, it will be of the form ≡ 1 (mod 6); and when a negative number is considered in Set B, it will be of the form ≡ 5 (mod 6).
So in the positive range 0<q; the probability of a number being prime in Set A is equal to p(A)-p(AB in A); and the probability of a number being prime in Set B is equal to p(B)-(p(AA in B)+p((BB in B)).
The probability of a number being composite in A or B when A or B is a negative number is the same as the probability of a number being composite in A or B when A or B is a positive number.
Thus the probability of P(A)-(P(AB in A)) is equal to P(-A)-((P(AA)+P(BB) in -A); and the probability of P(-B)-(P(AB in -B) is equal to P(B)-((P(AA)+P(BB) in B).
Since |A|=|B|, then P(A)-(P(AB in A)) is equal to P(B)-((P(AA)+P(BB) in B).
So the density twin primes relative to the probability of a prime occurring is equal to P(Twin Prime) = [P(A) – P(AB in A)] × [P(B) – P(AA+BB in B)] .
Preliminary Proof 1: P(AB) in A = P(AA+BB) in -A
Base Properties:
- All primes > 3 are form 6k±1
- A = {6x-1} ≡ 5 (mod 6)… -1 (mod 6)
- B = {6y+1} ≡ 1 (mod 6)
Product Forms:
- AB = (6x+1)(6y-1) = 36xy+6x-6y-1 ≡ 5 (mod 6) or… always -1 (mod 6), and let’s acknowledge (6x-1)(6y+1); which yields the exact same values, so we can just ignore it.
- AA = (6x-1)(6y-1) = 36xy-6x-6y+1 ≡ 1 (mod 6) or… always 1 (mod 6)
- BB = (6x+1)(6y+1) = 36xy+6x+6y+1 ≡ 1 (mod 6)… always 1 (mod 6)
Sign Change Properties:
- When k is in A, -k is in B
- When k is in B, -k is in A
- Negating AB products moves them from A to B
- Negating AA or BB products moves them from B to A
- Therefore -A is ≡ 1 (mod 6) and -B is ≡ 5 (mod 6)
Therefore:
AB composites in positive A = AA+BB composites in negative A
Preliminary Proof 2: |A| = |B| (Mirror Image)
Set Definitions:
- A = {…,-7,-1,5,11,…} ≡ 5 (mod 6) when A is positive (A) and ≡ 1 (mod 6) when A is negative (-A)
- B = {…,-5,1,7,13,…} ≡ 1 (mod 6) when B is positive (B) and ≡ 5 (mod 6) when B is negative (-B)
Bijective Mapping:
- For every k in A, -k exists in B
- For every k in B, -k exists in A
- No number can be in both A and B
Contradiction
Step 1: Definition of A and B
A = {6x-1} ≡ 5 (mod 6)
B = {6y+1} ≡ 1 (mod 6)
Step 2: Contradiction
If k ∈ A, then k ≡ 5 (mod 6)
If k ∈ B, then k ≡ 1 (mod 6)
Since 5 ≢ 1 (mod 6), we have a contradiction.
Conclusion
Therefore, our assumption that k belongs to both A and B is false.
Disjointness: A ∩ B = ∅
In other words, sets A and B are disjoint but have equal cardinality and are perfectly symmetrical.
Therefore: In any symmetric range [-q,q], |A| = |B|
Preliminary Proof 3: Sign Changes Preserve Composite Probabilities
For AB products:
If p is composite in A from AB, -p is composite in B
If p is composite in B from AB, -p is composite in A
For AA+BB products:
If p is composite in B from AA or BB, -p is composite in A
If p is composite in A from AA or BB, -p is composite in B
Therefore: Composite probabilities are preserved under sign changes
Preliminary Proof 4: P(AB) in A = P(AA+BB) in B for positive numbers
From Proof 1: P(AB) in +A = P(AA+BB) in -A
From Proof 2: |A| = |B| and sets are mirrors
From Proof 3: Sign changes preserve probabilities
Therefore: P(AB) in +A = P(AA+BB) in +B
Main Theorem Proof: Twin Prime Density
Given:
All previous proofs
Dirichlet’s theorem (infinite primes in A and B)
A and B are disjoint
For any k:
If 6k-1 is prime in A
And 6k+1 is prime in B
Then (6k-1, 6k+1) is a twin prime pair
Probability Analysis:
P(prime in A) = P(A) – P(AB in A)
P(prime in B) = P(B) – P(AA+BB in B)
Events are independent in A and B (disjoint due to different modulo classes)
Therefore: P(Twin Prime) = [P(A) – P(AB in A)] × [P(B) – P(AA+BB in B)]