Let $R_{p_i\#}$ be the reduced residue system relative the primorial for the $i$th prime.
Let $\left|R_{p_i\#}\right|$ be the number of elements in this set.
It is well known that:
$$\left|R_{p_i\#}\right| = (p_i-1)\cdot(p_{i-1}-1)\cdot(p_{i-2}-1)\cdot\,\dots\,\cdot(5-1)\cdot(3-1)$$
This is because each $x_j \in R_{p_{i-1}\#}$ forms a complete residue system relative $p_i$:
$$x_j,\, x_j+p_{i-1}\#,\, x_j+2p_{i-1}\#,\, \dots, \,x_j+(p_i-1)p_{i-1}$$
Here's my question.
What is the best lower bound estimate that I can make for counting the number of elements in their intersection?
$$\left|R_{p_{i-1}\#} \cap R_{p_{i}\#}\right|$$
It seems to me that this could often be less than $\left\lfloor\left|R_{p_{i-1}\#}\right|\dfrac{p_i-1}{p_i}\right\rfloor$ since we we are really asking how many $x_j \in R_{p_{i-1}\#} \equiv 0 \pmod {p_i}$
But I can't see a way to estimate this lower bound. If anyone could help or let me know the topic that helps me to figure this out, I would greatly appreciate it.
Thanks,
-Larry
Edit: I wrote a simple application to generate results up to $19$:
- $\left|R_{3\#} \cap R_{5\#}\right| = 1$, $\left|R_{3\#}\right|\cdot\dfrac{4}{5}=1.6$
- $\left|R_{5\#} \cap R_{7\#}\right| = 7$, $\left|R_{5\#}\right|\cdot\dfrac{6}{7}\thickapprox6.8571$
- $\left|R_{7\#} \cap R_{11\#}\right| = 43$, $\left|R_{7\#}\right|\cdot\dfrac{10}{11}\thickapprox43.636$
- $\left|R_{11\#} \cap R_{13\#}\right| = 443$, $\left|R_{11\#}\right|\cdot\dfrac{12}{13}\thickapprox443.0769$
- $\left|R_{13\#} \cap R_{17\#}\right| = 5420$, $\left|R_{13\#}\right|\cdot\dfrac{16}{17}\thickapprox5421.164$
- $\left|R_{17\#} \cap R_{19\#}\right| = 87307$, $\left|R_{17\#}\right|\cdot\dfrac{18}{19}\thickapprox87309.47368$
- $\left|R_{19\#} \cap R_{23\#}\right| = 1586754$, $\left|R_{19\#}\right|\cdot\dfrac{22}{23}\thickapprox1586754.7826$
- $\left|R_{23\#} \cap R_{29\#}\right| = 35236902$, $\left|R_{23\#}\right|\cdot\dfrac{28}{29}\thickapprox35236899.310$
I couldn't find this list on the OEIS.
Edit 2: I figured out conceptually but I'm still not clear the best way to approximate it mathematically.
$$\left|R_{p_i\#} \cap R_{p_{i+1}\#}\right| = \left|R_{p_i\#}\right| - \left| \left\{ x_j \in R_{p_{i}\#} \text{ and } x_j < \frac{p_{i}\#}{p_{i+1}} \right\} \right|$$
Edit 3: I think that I figured this out.
The answer should vary between $\dfrac{x-p_{i+1}+1}{p_{i+1}}$ and $\dfrac{x+p_{i+1}-1}{p_{i+1}}$
Am I right? I'll try to justify this in an answer if I don't hear any comments or an answer.
