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I am trying to prove the following conjecture:

Let it be $R_m(n) / m>0, m\in \Bbb N$ some reduced residue system modulo $n$ such that $R_m(n)$ is the reduced residue system between $(m-1)n$ and $mn$, and such that some element of the reduced residue system is a prime number.

Let it be $\lambda(R(n))$ the function expressing the number of consecutive reduced residue systems $R_m(n)$ (starting at $m=1$).

I conjecture that $\lambda(R(n))$ is always greater than $n$; that is, it exists some $R_m(n)$ for every $m \le n$.

In fact, I have checked that $\lambda(R(n))$ is much bigger than $n$, at least so it seems for $n \le 140$, as showed at this table I have calculated:

Values of $\lambda(R(n))$ for $n \le 140$

I would appreciate some advice and literature on the subject. Thanks in advance!

Edit: Below I have posted a graphic of $\lambda (R(n))$ with an approximation of the trend to an exponential function:

enter image description here

Juan Moreno
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    $f(n) = $ the least $m$ such that there is no prime in $[mn +1, mn+n]$. The probability that there is no prime in $[mn +1, mn+n]$ is about $(1-\frac{1}{\log mn})^n$. – reuns Nov 17 '18 at 06:19
  • Thanks for the comment @reuns. – Juan Moreno Nov 17 '18 at 14:01
  • Some idea regarding why it seems to be such a difference between the values of consecutive $\lambda R(n)$ values? For instance, $\lambda R(132) = 28985$, while $\lambda R(133) = 132722$. And it seems that such a dispersion is increasing for bigger values of $\lambda R(n)$ – Juan Moreno Nov 17 '18 at 14:09

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