# Questions tagged [primorial]

For problems involving either the product of the first n primes or the product of all primes up to and including n.

74 questions
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### Inequality involving prime numbers

If $p_k$ denotes the $k$ th prime then show that $$p_k \cdot p_{k+1} > p_{k+2}.$$  I think that Bertrand's pastulate and Bonse's inequality could be helpful but I exactly don't know how to use ...
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### Why is counting the number of least prime factors of a sequence of consecutive integers insufficient to resolve Legendre's Conjecture?

I've been thinking a long time about Legendre's Conjecture. A few nights ago, I came across the following argument which is of course too simple to be true. I would greatly appreciate if someone ...
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### Counting integers less than $n$ that are relatively prime to $x\#$

Let $x,n$ be integers such that $x < n$. Let $x\#$ be the primorial of $x$ so that $6\# = 5\# = 30$ and $7\# = 210$ Let gcd$(a,b)$ be the greatest common divisor of $a$ and $b$. Let $S(x,n)$ be ...
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### Is there an elementary argument for $\prod\limits_{p \le n}p < 3^n$ where $p$ is prime.

I was reading Hanson's proof that $\prod\limits_{p^a \le n}p^a < 3^n$ where $p$ is a prime and it occurred to me that there might be a simpler argument for $\prod\limits_{p \le n} p < 3^n$. Am ...
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### Show that $\lim_{n\to\infty}\mathcal I\left(\exp\left(\frac{2\pi\cdot i}{\log_n(p_n\#)}\right)\right)=0$

I'm pretty sure the imaginary part $\mathcal I$ of $f(n)=\exp\left(\frac{2\pi\cdot i}{\log_n(p_n\#)}\right)$ converges as $n\to\infty$, and probably to $0$. I stumbled upon this result studying the ...
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### Does c#k + i actually generate all primes above c# (plus some composites)?

The Wikipedia page for "Primality test", linked below , states that "all primes are of the form c#k + i for i < c# where c and k are integers, c# represents c primorial, and i represents the ...
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### A first encounter with the diophantine equations $P(x)=p_n\#$, where $P(X)\in\mathbb{Z}[X]$ has $\deg(P)=2$ and $p_n\#$ is a primorial: an example

In post I would like to ask about an example of a diophantine equation having the form $$P(x)=p_n\#\tag{1}$$ where $P(X)\in\mathbb{Z}[X]$ is a polynomial of degree $\deg(P)=2$ and $p_n\#$ denotes ...
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### Use pohlig-hellam algorithm to solve discrete log whose modulus is a primorial number

Recently I read a paper in which it use Pohlig-hellman algrithm to solve the follow formular: $$N \equiv 65537^c \pmod M$$ The target is to get c when N is given.The interesting thing is that the ...
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### Was in the literature the function $\sum_{n=1}^\infty\frac{1}{N_n^s}$, where $N_n$ is the $n$th primorial number?

I am curious to know if this function was in the literature, and mainly if it is possible define an analytic extension out of $\Re s>0$. Let $N_n$ the $n$th primorial number, and $s=\sigma+it$ ...
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### Smallest twin-prime of the form $k \cdot 11699$#$\pm 1$?

Denote $p$#$:=2\cdot 3\cdot 5\cdot 7\cdots p$ What is the smallest twin-prime of the form $k\cdot 11699$#$\pm 1$ , where $k$ is a positive integer ? Sieving out the candidates with Newpgen and ...
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### One particular limit $\lim_{k\to\infty} \frac{k}{p} = \infty$?

Let's take a value of $2^k$ and $p\#$, such that $2^k < p\#$ (but for the lowest possible $p$). As $p\#$ I understand primorial (product of the first $p$ primes). We have: ...
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### Asymptotic behaviour of Lambert series for Möbius or Euler's totient, evaluated at the reciprocal of a primorial

I denote for integers $n\geq 1$ the Möbius function with $\mu(n)$, and the Euler's totient function with $\varphi(n)$. Then inspired in Nicolas equivalence to the Riemann Hypothesis and the form of ...
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### What Is the Growth Rate of a Primorial?

As the title says, what is the growth rate of a primorial? I'm trying to evaluate the execution and data requirements of a function that uses one. Thanks!
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### Primorial asymptote

Is there any asymptotic representation for the primorial, and if it is there is it a consequence of the Riemann hypothesis or is it proved to be true without assumptions.
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### Equations involving primorials $N_k$: a reference request

I am asking this question exclusively as a reference request for equations similar than those that shows the section D25 of , but now involving primorials (see this MathWorld) instead of factorials....
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### Complexity $log(p_{n}\#)$

Is $log(p_{n}\#)$ a polynomial complexity (we count complexity after $n$)? $p_{n}\#$ is primorial. Could anyone prove it (if possible)?
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### For a given integers $x,n$, counting the number of integers $v$ where $x < v \le x+n$ and gcd$(\frac{n}{4}\#,v)=1$

If $n < 188$, is it true for all integers $x,n$ that there exist at least $4$ integers $v$ such that $x < v \le x+n$ and gcd$(v,\frac{n}{4}\#)=1$? I believe that the answer is yes. Here's my ...
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### How many examples like as this sum of distinct primes $2+3=5$ being $30$ a primorial, are there?

Since $2+3=5$ I know a simle example of a triple of coprime integers $(a,b,c)$, such that $$a+b=c,$$ being each summand a product of distinct primes $a=2$, $b=3$ and $c=5$, and their product is a ...
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### Is every primorial number squarefree?

Is every primorial number ( a number of the form $p$#$\pm 1$ ) squarefree ? According to my calculation, there is no prime $q\le 270,000$, such that $q^2$ can be a divisor of $p$#$-1$ or $p$#$+1$. ...
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### For which $k$ with $0<k<210637$ is $k\times 3571\# \pm 1$ a twin-prime-pair?

Because PARI/GP is not very fast in primilaty testing, I did not check the pairs $k \times 3571\# \pm 1$ in ascending order, but I begun with $k=200,000$ and got the twin prime pair 210637\times ...
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This program ...
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### Estimating the number of elements with a given least prime factor in a sequence of consecutive integers

Let $a,n$ be any positive integers. Let $\varphi(x)$ be the Euler totient function. It seems to me that the number of elements $x$ with $a \le x < a+n$ that have a given least prime factor will ...
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### Is there a standard notation for $(p_i-k)(p_{i-1}-k)(p_{i-2}-k)\cdots$ where $k$ is a small positive integer

For $k=0$, there is: $p_i\# = (p_i)(p_{i-1})(p_{i-2})\cdots(5)(3)(2)$ For $k=1$, there is: $\varphi(p_i\#) = (p_i-1)(p_{i-1}-1)(p_{i-2}-1)\cdots(5-1)(3-1)(2-1)$ Is there any other notation that ...
Let $x,y$ be positive integers and $p$ a prime. Is there a standard way to estimate the number of integers $z$ where $x \le z < x+y$ and $\gcd(z,p\#)=1$ For example, for $x=1000, y=30, p=7$, ...
Given $n\in\Bbb N$, we know that $\ln(n\#)\sim n$ where $n\#=\prod_{i=1}^{\pi(n)}p_i$ is primorial function with $p_i$ being $i$th prime while $\pi(n)$ being number of primes below $n$. How many ...