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..



Prove the existence of this number
This is almost surely false. The size of the largest gap between numbers coprime to $N=N\_p$ grows at a faster rate than do the primes. There may even be an example with q less than 1000 where q is the least prime factor of the numbers c and d, and every number in between c and d has a smaller least prime factor. Such c and d would witness no beta(j) of your conjecture.


**Update 2019.08.11** At <https://mathoverflow.net/a/115879> Aaron Meyerowitz has an example with two integers with least prime factor 73, and all integers in between having smaller least prime factor. This is my (or Aaron's) proffered counterexample. I conjecture that this is the smallest such, and would be glad to be shown otherwise.


**End Update 2019.08.11.**


Gerhard "There May Be Smaller Examples" Paseman, 2019.08.09.


