# Prove the existence of this number

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..

• "There exist at least $m$ in $\bf N$" doesn't parse. Aug 3, 2019 at 23:41
• “at least $m$ integers” would be the more natural way to say that. Aug 4, 2019 at 2:34
• I reformulate the sentence Aug 4, 2019 at 9:39
• Should word "one" replace "an" in the phrase "at least an integer $m$", so that the correct phrase would be: "at least one integer $m$"? Aug 9, 2019 at 22:23

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.
• how can you prove your claim? The Jacobsthal function (of primorial numbers $h(n)$) for example can't help in this case, because the size of the interval $]q \beta_p(i), q \beta_p(i+1)[$ change.. we know that if the size of the interval is always $2q$ my claim is false ($h(n)$ with $q=p_n$ can take values large than $2q$) Aug 10, 2019 at 19:02
• What is your counter-example ? and i don't know a computer can verify up to $\displaystyle{\small \prod_{\substack{a \leq q \\ \text{a prime}}} {\normalsize a}} = e^{q+o(1)}$ with $q$ near $1000$ Aug 10, 2019 at 22:05