Primes in reduced residue systems
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I am trying to prove the following conjecture:
Let it be $R_m(n) / m>0, min Bbb N$ 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:
prime-numbers reduced-residue-system
asked Nov 14 at 16:21
Juan Moreno
163
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up vote
2
down vote
favorite
I am trying to prove the following conjecture:
Let it be $R_m(n) / m>0, min Bbb N$ 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:
prime-numbers reduced-residue-system
asked Nov 14 at 16:21
Juan Moreno
163
$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 at 6:19
Thanks for the comment @reuns.
– Juan Moreno
Nov 17 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 at 14:09
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I am trying to prove the following conjecture:
Let it be $R_m(n) / m>0, min Bbb N$ 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:
prime-numbers reduced-residue-system
asked Nov 14 at 16:21
Juan Moreno
163
I am trying to prove the following conjecture:
Let it be $R_m(n) / m>0, min Bbb N$ 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:
prime-numbers reduced-residue-system
prime-numbers reduced-residue-system
asked Nov 14 at 16:21
Juan Moreno
163
asked Nov 14 at 16:21
Juan Moreno
163
edited Nov 18 at 23:05
asked Nov 14 at 16:21
Juan Moreno
163
asked Nov 14 at 16:21
Juan Moreno
163
asked Nov 14 at 16:21
Juan Moreno
163
163
$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 at 6:19
Thanks for the comment @reuns.
– Juan Moreno
Nov 17 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 at 14:09
add a comment |
$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 at 6:19
Thanks for the comment @reuns.
– Juan Moreno
Nov 17 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 at 14:09
$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 at 6:19
$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 at 6:19
Thanks for the comment @reuns.
– Juan Moreno
Nov 17 at 14:01
Thanks for the comment @reuns.
– Juan Moreno
Nov 17 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 at 14:09
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 at 14:09
add a comment |
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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 at 6:19
Thanks for the comment @reuns.
– Juan Moreno
Nov 17 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 at 14:09