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I analyzed primes generated by applying four linear offsets to the first element of twin prime pairs ( 𝑝 , 𝑝 + 2 ), for 𝑝 ≀ 1,000,000:

π‘ž1 = 2𝑝 + 1, π‘ž2 = 2𝑝 + 7, π‘ž3 = 2𝑝 βˆ’ 3, π‘ž4 = 2𝑝 + 3.

About 57% of pairs produce at least one prime among these offsets. The rate varies by last digits of the twin primes: pairs ending in (1,3) and (7,9) have ~64% success, but those ending in (9,1) only ~43%.

I’d appreciate feedback on whether these patterns are known, and suggestions for related primes or offsets to investigate. Code and data are available here: https://github.com/allen-proxmire/twin-primes-offsets

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    $\begingroup$ It may be interesting to separate categories by position modulo a primorial, e.g., $6,30,210,2310,\dots$. Obviously there are effectively $100$% of twin prime pairs present in the single category modulo $6$, and I would expect that pairs are relatively evenly divided among the three categories modulo $30.$ $\endgroup$ Commented Aug 11 at 22:17
  • $\begingroup$ great idea. I haven't looked at mods yet. $\endgroup$ Commented Aug 12 at 15:30

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In the (1,3) case we have:
$p \equiv 1 \mod 10 => q_4 \equiv 2 p + 3 \equiv 5 \mod 10$
So $q_4$ is never prime. While testing, I found that $q_1, q_2$ and $q_3$ produce primes.

In the (7,9) case we have:
$p \equiv 7 \mod 10 => q_1 \equiv 2 p + 1 \equiv 5 \mod 10$
So $q_1$ is never prime in this case. $q_2, q_3$ and $q_4$ produce primes.

In the (9,1) case we have:
$p \equiv 9 \mod 10 => q_2 \equiv 2 p + 7 \equiv 5 \mod 10$
$p \equiv 9 \mod 10 => q_3 \equiv 2 p - 3 \equiv 5 \mod 10$

So $q_2$ and $q_3$ are never prime in this case. $q_1$ and $q_4$ produce primes.

Since I found no restrictions for the other cases, in the long run I would suspect that (1,3) and (7,9) each produce 50% more primes among $q_1, q_2, q_3, q_4$ than (9,1).
This aligns pretty well with the 43% and 64%

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  • $\begingroup$ I'm my GitHub you can find these results in a similar fashion. I breakdown how many primes are found by the four offsets. as Michael Stocker says, at least one offsets produces no primes in each of the last-digit-categories. initially I applied only two offsets to each group, but had a total of four different ones. applying all four, I found a few extra primes. thx $\endgroup$ Commented Aug 13 at 13:20

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