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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.

2
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2answers
88 views

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 ...
12
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0answers
226 views

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 ...
4
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2answers
165 views

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 ...
10
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0answers
223 views

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 ...
0
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1answer
64 views

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 ...
0
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0answers
28 views

Does c#k + i actually generate all primes above c# (plus some composites)?

The Wikipedia page for "Primality test", linked below [1], 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 ...
1
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0answers
40 views

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 ...
1
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1answer
116 views

What's the sum of the inverses of the Primorial numbers?

What's the sum of the inverses of the primorial numbers? Let the $n^{th}$ primorial number be the product of the first $n$ primes $\displaystyle n\#= \prod_{p\leq p_n}p$ So $N\#=2,2\cdot3,2\cdot3\...
3
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2answers
105 views

Sum of reciprocals of primorials

It is well known that the sum $$ \sum _{{k=0}}^{\infty }{\frac {x^{k}}{k!}} $$ converges to $e^{x}$. In particular, for $x=1$ we have $\sum _{{k=0}}^{\infty }{\frac {1}{k!}}=e$. But what about the ...
2
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0answers
54 views

On numbers of the form $\pm x^2\pm N_k$, with $x\geq 1$ integer and $N_k$ denoting primorials

In this post we consider positive integers $a$ of the form $$\pm x^2\pm N_k\tag{1}$$ with $x\geq 1$ integer and for integers $k\geq 1$ denoting $$N_k=\prod_{j=1}^k p_j$$ as the primorial of order $k$,...
0
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1answer
78 views

What about of $\sum_{n=1}^\infty 1/10^{N_n}$ as transcendental number?

In this post we denote the primorial of order $n$ as $N_n=\prod_{k=1}^n p_k,$ where thus $p_k$ denotes the $k$th prime number. This section of the Wikipedia's article dedicated to transcendental ...
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0answers
40 views

|$q^2$-$p*r$|=prime(n)#

With three consecutive primes p,q,r |$q^2$-$p*r$|=prime(n)#. There are two solutions of 7,11,13 and 17,19,23 for 3#=30 and two solutions of 97,101,103 and 107,109,113 for 4#=210. Do you think there ...
3
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2answers
78 views

How many zeroes end 2718#? [Primorial Prime Topic]

The primorial of n, written n#, is the product of the prime numbers less than or equal to n. I cannot find anywhere how to determine the answer to finding the number of trailing zeroes for 2718#. I ...
2
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0answers
103 views

A conjecture about the asymptotic of the absolute value of $\sum_{k=1}^n\sin\left(N_k\right)$, where $N_k$ is the primorial of order $k$

I would like to ask this question to know if it is possible to say something about the asymptotic of next sequence, that seems erratic. I was inspired in a sequence showed in this post of Mathematics ...
12
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1answer
315 views

On the conjecture that, for every $n$, $\lfloor e^{\frac{p_{n^2}\#}{p_{n^2 + 1}}}\rfloor $ is a square number.

I was playing around with numbers and I discovered that $$\left\lfloor e^{\frac{p_4\#}{p_5}}\right\rfloor=\left\lfloor e^{\frac{210}{11}}\right\rfloor =13981^2,$$ with floor function $\lfloor x \...
0
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1answer
98 views

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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0answers
48 views

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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0answers
56 views

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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1answer
64 views

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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0answers
53 views

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 ...
0
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0answers
106 views

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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0answers
48 views

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.
3
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0answers
45 views

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 [1], but now involving primorials (see this MathWorld) instead of factorials....
0
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1answer
35 views

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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0answers
35 views

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 ...
5
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1answer
82 views

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 ...
2
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0answers
117 views

Finding large prime gaps without using brute force

It seems to me that there are three ways to find large prime gaps without using brute force. Natural Prime Order Using Factorials For example, using $7!$, it is clear that $(7!+2), (7!+3), \dots (7!+...
1
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0answers
105 views

What is a fast algorithm to compute the primorial $P_{n}$ given the primorial $P_{n+1}$?

This question is related to my other question regarding entropy with respect to a given multiplicative function ( Limit for entropy of prime powers defined by multiplicative arithmetic function ). ...
1
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1answer
208 views

Estimating the number of integers less than $m$ that are relatively prime to $p_n\#$

Let $m \ge 2$ be an integer. Let $p_n$ be the $n$th prime so that $p_1 = 2, p_2 = 3,$ etc. Let $p_n\#$ be the primorial for $p_n$. Let $\gcd(a,b)$ be the greatest common divisor for $a$ and $b$. ...
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0answers
63 views

Infinite sum of the reciprocal of “anti-primorials”

Multiple questions have been submitted regarding the sum of the reciprocal of primorials, i.e. $$ S_P=\sum_{n=1}^{\infty}P_n = \frac{1}{2} + \frac{1}{2\cdot3} + \frac{1}{2\cdot3\cdot5} + \ldots \...
12
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2answers
985 views

A question about numbers from Euclid's proof of infinitude of primes

Observe this list: $$ \begin{aligned} 2+1&=3\\ 2\cdot3+1&=7\\ 2\cdot3\cdot5+1&=31\\ 2\cdot3\cdot5\cdot7+1&=211\\ 2\cdot3\cdot5\cdot7\cdot11+1&=2311\\ 2\cdot3\cdot5\cdot7\cdot11\...
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0answers
30 views

Counting the number of integers $1 < i < p\#$ where gpf($i$) > $p$

lpf = least prime factor gpf = greatest prime factor $p\#$ = the primorial of the prime $p$ It is straight forward to count the number of integers $i$ where $1 < i < p\#$ and lpf($i$) $> p$. ...
7
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0answers
252 views

What's infinte sum of the reciprocal of the primorial?

$$\sum_{n=1}^\infty \frac{1}{p_n\#} = \frac{1}{2}+\frac{1}{2\times3}+\frac{1}{2\times3\times5}+\dots$$ where $p_n\#$ is the nth Primorial. Does this sum approaches some known value or constant and ...
3
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0answers
164 views

On a combination between an equivalence to the Riemann's Hypothesis and the abc conjecture

If my calculations were rights, it's easy to deduce combining the abc conjecture with Nicolas equivalence to the Riemann's Hypothesis that: $\qquad\qquad\forall \epsilon>0$ there exists a positive ...
1
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1answer
141 views

Is this observation equivalent to Legendre's conjecture? $\forall n \in \Bbb N^+, \exists k \in \Bbb N^+:\ \ (n^2)\# \lt k! \lt (n+1)^2\#$

Is this observation equivalent to Legendre's conjecture? $$E1:\forall k \in \Bbb N^+ \not \exists n,m \in \Bbb N^+, n \not = m:\ \ \ k! \le n^2\# \lt m^2\# \le (k+1)! $$ This observation is a ...
4
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1answer
171 views

Is there an equivalent to the Bertrand's postulate between factorials and primorials?

As the title explains, I am trying to know if there is a definition about the upper limit to find the first primorial $p_i\#$ (following the definition at OEIS) existing after a given factorial $n!$ (...
2
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1answer
151 views

Prime conjecture containing primorial: the difference between the primorial $n\#$ and the smallest prime $p > n\# + 1$ is always a prime

Help me find the exact conjecture statement. What I roughly remember is that it stated that the difference between primorial $n\#$ (product of first $n$ primes) and "some" larger number than the ...
4
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1answer
54 views

Prove that for $n \ge 3$, for any integer $w$, $\exists x$ such that $w < x < w + p_n\#$ and lpf$(x) \ge p_{n+3}$

Let $p_n$ be the $n$th prime where $n \ge 3$. Let $p_n\#$ be the primorial for $p_n$ and let lpf$(x)$ be the least prime factor for $x$. I am feeling like this should be very easy to prove but I am ...
0
votes
1answer
128 views

Asymptotic bounds on sum of primes

Let $p_i$ denote the $i$th prime number, and let $p_k\#$ denote the $k$th primorial, $p_k\# \overset{\textrm{def}}= \prod_{i \le k} p_i$. I am interested in asymptotic upper bounds for the function ...
1
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0answers
59 views

Is there a standard mathematical way to analyze the gaps between elements of a reduced residue system modulo a primorial?

I've been very interested at the gaps the between the elements of a reduced residue system modulo a primorial $p\#$. The reason for this interest is that unlike the primes, the elements of reduced ...
3
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1answer
57 views

Would this sequence (OEIS A068374) be somehow attached to the twin prime conjecture?

Today I came across an interesting sequence at OEIS, A068374, described as "Primes $n$ such that positive values of $n$-Primorial($k$) are all primes ($k\gt0$)". The sequence is as follows: $(2, 5, ...
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0answers
106 views

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 $p$#$+1$. ...
1
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0answers
29 views

Reasoning about the congruence classes modulo $p_n\#$ related to $(\frac{p_n\#}{p_i})^k$

Let $p_n$ be the $n$th prime and $p_n\#$ the primorial for $p_n$. Is it correct to assume that for any $i \le n$ and $w$, there exists $a < p_i$ such that: $$\left(\frac{p_n\#}{p_i}\right)^w \...
1
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0answers
64 views

Reasoning about Chinese Remainder Theorem relative a primorial where for all primes some number $k \equiv 1 \pmod {p_i}$

Let $p_1, p_2, \dots, p_n$ be the first $n$ primes and $p_n\#$ be the primorial of $p_n$ Using the Chinese Remainder Theorem and Fermat's Little Theorem, the following, I believe, is true: $$\sum_{i}...
3
votes
2answers
84 views

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 ...
1
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1answer
85 views

Does $\log(p \#)\approx p$ hold for large $p$?

This program ...
2
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0answers
59 views

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 ...
1
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0answers
27 views

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 ...
0
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0answers
63 views

Estimating the number of integers in a sequence of consecutive integers that are relatively prime to a given primorial

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 $x=1000, y=30, p=7$, ...
0
votes
1answer
149 views

Factors of primorial of number

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 ...