I started with this question about linking at least one twin prime with each prime. My approach is:
- Give each prime two buckets.
- Put each odd 3n in the second bucket of it's largest prime divisor.
- Put it in the first bucket if (3n-4, 3n-2) are twin primes.
- Prove that no first bucket is empty, or that the density drops slow enough.
Edit: Let
$d(p, 3n) =
\begin{cases}
1 & p|n \wedge p \text{ is largest prime divisor}\\
0 & \text{ otherwise }
\end{cases} \\
t(p, 3n) =
\begin{cases}
1 & d(p, n) = 1 \wedge (3n-4, 3n-2) \text{ are twin primes}\\
0 & \text{ otherwise }
\end{cases} \\
\\
f(p) = \sum_{3n}^N {t(p, 3n)} \\
g(p) = \sum_{3n}^N {d(p, 3n)} \\
h(p) = \frac {f(p)}{g(p)} \\
$
When I graph $f(p)$, it's about what you'd expect:
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And the graph of $g(p)$ is no surprise either:
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However, when I graph $h(p)$ for $N = 20M, 30M, ..$ I get this:
(y = 5%, 10%, 15%; x = primes < 20000; color = N +10M)
Ratio of 3pn twins to total 3pn
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Are these lines a result of my small sample size and ratio-similarity? Or is this a pattern in how twin primes are distributed?