I've been thinking the primorial for say the $i$th prime $p_i$and the equations for counting the number of elements in the reduced residue system for this primorial and counting the number of elements of the form $6x-1$ that have a sister element of $6x-1+2$
Let $R$ = the set of elements of the reduced residue system for $p_i\#$
Let $T$ = the set of elements of the reduced residue system of form $6x-1$ where $(6x-1)+2$ is also an element of $R$.
Let $[R]$ = the number of elements of $R$ and $[T]$ = the number of elements of $T$
I know that these equations are well known:
- $|R|= (p_i-1)\cdot(p_{i-1}-1)\cdot(p_{i-2}-1)\cdot\,\dots\, \cdot(3-1)$
- $|T|= (p_i-2)\cdot(p_{i-1}-2)\cdot(p_{i-2}-2)\cdot\,\dots\, \cdot(5-2)$
Here are my questions:
- If we define $S$ as the number of elements that do not have a twin element, is it correct that: $|S| = |R| - 2\cdot|T|$
- If $\pi(x)$ is the prime counting function. Is it correct that the twin prime conjecture would be proved if it turned out that:
$$\pi(p_i\#) > |S| + \dfrac{1}{2}|T|$$
- Assuming that my reasoning is correct, how close have folks gotten to proving this comparison? Is there any paper that you can cite about the twin prime conjecture?
Thanks,
-Larry
Edit: I did some calculations and here's what I found:
- For $p_i = 5$, $\pi(p_i\#) - |S| + \dfrac{1}{2}|T| = 4.5$
- For $p_i = 7$, $\pi(p_i\#) - |S| + \dfrac{1}{2}|T| = 16.5$
- For $p_i = 11$, $\pi(p_i\#) - |S| + \dfrac{1}{2}|T| = 61.5$
But then:
- For $p_i = 13$, $\pi(p_i\#) - |S| + \dfrac{1}{2}|T| = -290.5$
- For $p_i = 17$, $\pi(p_i\#) - |S| + \dfrac{1}{2}|T| = -16422.5$
Edit: Can anyone prove that if $p_i > 11$, then:
$$\pi(p_i\#) < |S| + \dfrac{1}{2}|T|$$