Local Structure of Prime Numbers in Clean Blocks

Description and goal (extract from the paper)

We consider blocks of 41 consecutive integers, \(B_m = [41m+1, 41m+41]\), \(m\ge 0\).

We introduce the notion of a 3,5,7–clean block: every odd composite number \(> 1\) in such a block has a divisor \(3\), \(5\) or \(7\).

For such a block we define candidates after sieving by \(3,5,7\) (among odd numbers \(>7\)) as those integers which are coprime with \(M = 3\cdot 5\cdot 7 = 105\), i.e.\ \(\gcd(n,105)=1\) and \(n\) is odd.

Local lemma: In every 3,5,7–clean block all numbers \(n\) satisfying \[ n\in B_m,\quad n>7,\quad n\ \text{odd},\quad \gcd(n,105)=1 \] are prime numbers. In every block of length \(41\) there are \(8\)–\(10\) such candidates, which implies that in a clean block there are at least \(8\) primes \(>7\).

2. Key Mathematical Formulas ➗

Formulas defining the sieve and candidates

Base modulus of the sieve: \[ M = 3 \cdot 5 \cdot 7 = 105. \]

Candidates (in the sense of the paper): \[ n \in S(B_m)\ \Longleftrightarrow\ n\in B_m,\ n>7,\ n\equiv 1\pmod 2,\ \gcd(n,105)=1. \]

The full pattern of candidates (including parity) is periodic modulo \(\operatorname{lcm}(2,105)=210\), i.e. on an interval of length \(210\).

3,5,7–clean block and local lemma

A 3,5,7–clean block (\(B_m\)): \[ \forall n\in B_m,\ \text{if } n>1\ \text{is odd and composite, then }\ 3\mid n\ \lor\ 5\mid n\ \lor\ 7\mid n. \]

Local lemma: \[ \text{If } B_m \text{ is 3,5,7–clean, then every } n\in S(B_m) \text{ is a prime number.} \]

Constructive local proof of the existence of twin pairs

A twin pair of candidates in a block: \[ \exists n\;\bigl(\{n,n+2\}\subseteq B_m\ \land\ n\in S(B_m)\ \land\ n+2\in S(B_m)\bigr). \]

Analysis of the full pattern (two cycles modulo \(105\), total length \(210\)) shows that the largest gap between consecutive starts of twin candidate pairs is \[ G = 30. \] Hence for block length \(D=41\) we have \[ 41 > 30 = G, \] which guarantees the existence of at least one twin pair of candidates in every block of length \(41\), and in a 3,5,7–clean block – a pair of twin prime numbers.

3. Analysis of a Block of Length 41

4. Interactive Sieve Visualization (two cycles modulo 105)

Shift the block (start on the axis of two cycles modulo 105, full pattern of length 210):

Block start (mod 105): 1 | Block end (mod 105): 41 | Length: 41

Current start with respect to the full cycle of length 210 (mod 210): 1, number of twin pairs for this shift: 0

Global statistics (full cycle of length 210)

Minimal number of twin candidate pairs in a block of length 41 (over all 210 shifts): ?

Maximal number of twin candidate pairs: ?

Shifts (mod 210) where the minimum is attained:

Array of the number of pairs for all shifts (starts 0..209, block [start, start+40]):

Yellow cells: Twin pair (e.g. 11, 13). Green cells: Other candidate (not part of a twin). Grey cells: Composite/even number (removed by the sieve 3, 5, 7).

1. Summary of the Paper and Definitions 📝