Prime Number Congruence Conjecture

Question

$6±1 = (5,7)$

5 and 7 are twin primes

$6 + 5 = 11$

$6 + 7 = 13$

11 and 13 are twin primes

$ 6 × 5 = 30 $

$ 6 × 7 = 42 $

30 and 42 are adjacent to the twin primes (29. 31) and (41, 43)

Is there any other multiple of 6 that is between twin primes (5,7), produces this same pattern of primes in the sums (11, 13) and primes adjacent to its products (29. 31) and (41, 43)? If so, how often does it occur?

Disclaimer: I don't know the mathematics necessary to figure it out on my own. I'm asking out of innocent curiosity.

Answer 1

I've found two numbers that fit your requirements exactly - 6 and 109505970 - but if we loosen them a bit then we get an interesting sequence! The numbers we're trying to find need to satisfy the following conditions. If $n$ fits the pattern, then:

• $n+(n-1)$ is prime
• $n+(n+1)$ is prime
• $n(n-1) - 1$ is prime
• $n(n-1) + 1$ is prime
• $n(n+1) - 1$ is prime
• $n(n+1) + 1$ is prime

Below 10 million, there are 25 numbers that fit this pattern: $3$, $6$, $21$, $1365$, $86604$, $185535$, $411501$, $759759$, $833799$, $1192290$, $1297815$, $2092719$, $2130324$, $2876160$, $3469311$, $3515799$, $5268606$, $5335959$, $7279791$, $7544901$, $7749435$, $7787661$, $7994085$, $8067501$, and $9954141$. This sequence ignores the "multiple of 6" condition and the "$n-1$ and $n+1$ have to be twin primes" condition.

If you add the condition that $n-1$ and $n+1$ also have to be prime, then the only numbers that I've found that satisfy this condition are $6$ and $109505970$, and $109505970$ is actually a multiple of 6!