Let $p,q \in \mathbb{P}$, $p \geq 3$ and $q$ is the next prime to $p$.

For $b \in \mathbb{P}$ Consider : $N_b = \displaystyle{\small \prod_{\substack{a \leq b \\ \text{a prime}}} {\normalsize a}}$

Let $n \in \mathbb{N}$,

$n$ is **q-point** iff $n = q \alpha$ with $\gcd(\alpha, N_p)=1$

My Conjecture:There is at least one integer $m$, between two consecutive q-points in which $\gcd(m, N_q)=1$

We can reduce this conjecture to the following problem:

For $b \in \mathbb{P}$ We have :

$$\#\{k \leq N_b \, | \, \gcd(k, N_b)=1\} = \displaystyle{\small \prod_{\substack{a \leq b \\ \text{a prime}}} {\normalsize (a-1)}}$$

Let $\beta_b(i)$ be the i-th number coprime to $N_b$ and less than $N_b$, $i \in \{1,2,3,\cdots,\displaystyle{\small \prod_{\substack{a \leq b \\ \text{a prime}}} {\normalsize (a-1)}}\}$

My Conjecture:$(\forall i \in [1,\displaystyle{\small \prod_{\substack{a \leq p \\ \text{a prime}}} {\normalsize (a-1)}}-1]) (\exists j \in [1,\displaystyle{\small \prod_{\substack{a \leq q \\ \text{a prime}}} {\normalsize (a-1)}}])$ : $$q \cdot \beta_p(i) < \beta_q(j) < q \cdot \beta_p(i+1)$$

I had hard feeling that this conjecture is true, but all my Attempts to prove it failed.

Many thanks for any help..

one" replace "an" in the phrase "at least an integer $m$", so that the correct phrase would be: "at least one integer $m$"? $\endgroup$ – Wlod AA Aug 9 '19 at 22:23