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.
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Is the product of primes from $n/2$ to $n$ always less than $2^n$?
Calculating for the first few $n$ and assuming that $p_{i}$ is the $i$th prime, it appears that $\left( \prod_{(\frac{n}{2} \lt p_{i} \le n)} p_{i} \right) \lt 2^{n}$ .
For example, $\left( \prod_{(\...
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Is there a more efficient way to find the least prime factor?
Assuming $Q_{k} \equiv p_{k}\text{#} + 1$, my goal is to find the least prime factor of $Q_{k}$ for each integer $k = 1 \ldots 100$ . The Python program shown below tries using SymPy to do so, but ...
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For which n is p(n)# + 1 prime? [duplicate]
The primorial $\color{blue}{p_{11}\text{#} = 200560490130}$, and $\color{blue}{p_{11}\text{#} + 1 = 200560490131}$ is prime. There are other times when $\color{blue}{p_{n}\text{#} + 1}$ is not prime; ...
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Proof that the $(n+1)$-th prime is less than or equal to the $n$-th primorial.
Assuming that $p_{n}$ is the $n^{ th}$ prime and $p_{n}\text{#}$ is the $n^{ th}$ primorial, what is a proof in elementary number theory that, for all $n \ge 2$, $p_{n+1} \le p_{n}\text{#}$ ?
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How do you multiply in Primorial number system?
Primorial number system is a number sytem that uses primorials which are defined as follows :
Let $p_1=2, p_2=3,p_3=5,p_4=7,p_5=11,...$ the primes. The sequence of primorials, noted $p_n\#$ is $$(p_n\#...
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Help to mathematically theorize the eventual value of Primorial number system
The primorial numeral system is a numeral system whose interest I am trying to establish mathematically.
Unfortunately, my mathematical knowledge is limited, so I would like to briefly present some ...
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Examples of illustrations of a mathematical object that is a priori correctly constructed
Following Jean Pierre Serre's way of introducing $p$-adic numbers in his Cours d'arithmétique (1970), referring to the footnotes of the post for some possibly useful details :
$\forall n\geq 1$, let $$...
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How do you find constants in an k-Tuple conjectures?
By introducing modular objects associated to the sequel of rings $$(Z/2,Z/6,Z/30,Z/210,Z/2310,..., Z/p_n\#Z)$$
a sequence of coefficients is updated$$(2;\color{green}{\frac83}; 3.2;...$$
(see my ...
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Difficulties with $\mathbb Z_3$, $3$-adic numbers and a relative questioning
I'm reading Jean-Pierre Serre's 1970 "Cours d'arithmetique". I'm having trouble reading the beginning of his chapter 2 devoted for example to $\mathbb Z_3$, $3$-adic numbers that I discover ...
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Estimating the upper bound for $\prod\limits_{p \le x}{p^{\frac{1}{p}}}$
An upper bound for the primorial can be found based on the first chebyshev function.
From $\vartheta(x) < 1.00028x$, it is clear that:
$$\prod\limits_{p \le x}p \le e^{1.00028x} < (2.72)^x$$
I ...
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A result about $\mathbb Z/p_n\#\mathbb Z$ that I want to prove
I claimed in a recent post that
Let $p_1=2,p_2=3,p_3,...$ the prime numbers. $$\forall n \in \mathbb N^*,p_n\#:=p_1...p_n\text{ (n-th primorial)}$$ $$U_{p_n\#}:=(\mathbb Z/p_n\#\mathbb Z)^\times$$ $$...
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Questions about $\mathbb Z/30\mathbb Z$
I'm interested in the ring $(\mathbb Z/30\mathbb Z,+,\times)$, the elements of which I'll write down in bold. For example $$\textbf{7}=7+30\mathbb Z=\{...-83,-53,-23,7,37,67...\}.$$
I'm interested in ...
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A new topology on $\Bbb{Z}$ based upon basis $X_d = $ the insolubility of $x^2 = 1\pmod d$ etc
Let $p_n$ denote the $n$th prime throughout and define $p_n\# := p_n p_{n-1} \cdots p_1$ to be the $n$th "primorial".
Define the topology $\tau$ on $\Bbb{Z}/p_n\#$ to be that generated by ...
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Question related to the primorial of the 1000th prime number , 7919#.
Suppose $m=7919$# which is the product of the first $1000$ prime numbers.
It is easy to find primes of the form $m\cdot n+1$ , where $n$ is a positive integer.
But what is the smallest positive ...
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Sum of reciprocals of primorial numbers
What is the following sum?
$x=\frac{1}{2}+\frac{1}{2\times 3}+\frac{1}{2\times 3\times 5}+...$
$=\Sigma_{n=0}^{\infty} \frac{1}{\prod_{i=1}^{n}p_i}$, where $p_i$ is the $i^{th}$ prime.
I'm guessing ...
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The function $f(n) = \sum_{d \mid p_n\#} \mu(d)\sum_{r^2 = 1\mod d}\lfloor\frac{p_{n+1}^2 - 2 - r}{d}\rfloor$ has no fixed point $f(n) = n$?
Definition.
$$
f(n) := \sum_{d \mid p_n\#} \mu(d)\sum_{r^2 = 1\mod d}\lfloor\frac{p_{n+1}^2 - 2 - r}{d}\rfloor
$$
where $p_{n+1}$ is the $(n+1)$th prime number. And where it is understood that each ...
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What relationship does the primality characteristic function have with the summation group that it's over? Can we form a sheaf?
Let $G_n$ be the finite group of square-free integers that are divisors of $p_n\#$ under the law $x \star y := \dfrac{ x y}{\gcd(x,y)^2}$. It obviously forms a boolean group. Where $p_n\#$ is $p_2 ...
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Continued aliquot sums
What happens if one takes the aliquot sum of an integer and then repeats the process so that one takes the aliquot sum of all of those factors that were not reduced to the number 1 on the previous ...
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Finding an $x$ that satisfies the Jacobsthal function for the primorial of the $n$th prime.
The Online Encyclopedia of Integer Sequences has a nice page on the gaps found related to the Jacobsthal function for primorials.
I would like to take a look at the sequences that have these gaps ...
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Calculating the nth totative of a large primorial
For finding the $n$th prime (or the number of primes less than a number), there are results for large numbers; I was wondering if something similar for the totatives of a primorial number is possible (...
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Growth of x relative to the Primorials?
I'm trying to find out the growth rate of an algorithm that grows with $p_{n}\#$ where $p_{n}\#$ is the smallest primorial that is larger than the input: $x\in\mathbb{Z^{+}}$.
Other research has ...
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Proving $x*\ln(\ln(x))>\ln(\ln(x\#))$ for $\infty>$x>e$
Let $x \in \mathbb{N}$.
For $\infty>x>e$,
Let $x$ be the largest prime in the factorization of the primorial $x\#$.
How does one go about proving $x*\ln(\ln(x))>\ln(\ln(x\#))$?
The plot of ...
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$H_{c,n}(y)=\sum_{c\mid d\mid p_n\#}(-1)^{\omega(d)}[\dfrac{y-z_{c,d}}{d}]$ is asymptotic to $\sum_{c\mid d\mid p_n\#}(-1)^{\omega(d)}[\dfrac{y}{d}]$?
For $c, n \in \Bbb{N}$, define the function $H_{c,n}(y) = \sum_{c \mid d \mid p_n\# \\ d\leq y} (-1)^{\omega(d)} \lfloor \dfrac{y - z_{c,d}}{d}\rfloor$.
Where $z_{c,d}$ is a random variable in the ...
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Reasoning about limits related to the prime number theorem
From the Prime Number Theorem, it follows that:
$$\lim\limits_{n \rightarrow \infty}\sqrt[n]{n\#} = e$$
One of the standard definitions of $e$ as found here is that:
$$e = \lim\limits_{n \rightarrow \...
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Does it follow that $\lim\limits_{n \rightarrow \infty} \sqrt[n]{n\#} = e$?
In the Wikipedia article on primorial, in the Characteristics Section, it is stated:
$$\lim\limits_{n \rightarrow \infty} \sqrt[n]{n\#} = e$$
This sounds reasonable to me. There is a footnote that ...
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Does Chebyshev's theorem provide a lower bound of the primorial $n\#$ such that $n\# \ge 2^{n/2}$
I found the following claim here:
Chebyshev's theorem gives the lower bound $2^{(n/2)}$.
Is this correct?
If $n\#$ is the primorial of $n$, does it follow that:
$$n\# \ge 2^{(n/2)}$$
As I understand ...
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Tight lower and upper estimate of alternating summation of reciprocals of divisors of primorial $\sum_{d \mid p_n\#} (-1)^{\omega(d)} \dfrac{1}{d}$?
Let $f(n ) = \sum_{d \mid p_n\#}(-1)^{\omega(d)} \dfrac{1}{d}$ where $p_n$ is the $n$th prime number and $p_n\# = p_n p_{n-1} \cdots p_1$.
I'm looking for a way to make a simple estimate of (upper / ...
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Expressing any even natural number as a sum of primorials with coefficients
I'm having a hard time trying to solve the following problem:
Given any random even natural number, $x$, prove that it can or cannot be written as the product of some integer, $b$, times the primorial ...
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Stuck on reading a proof of $n\# < 4^n$
Hey guys so I am reading through a proof and I am completely stuck on one part.
The lemma is $n\# < 4^n$ where $n\#$ is the primorial function.
So far I have understood a bit of the proof and it ...
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Why does the golden ratio emerge in this primorial-related sequence?
Let $$f(i):=\left\lfloor\frac{p_i\#}{\varphi(p_i\#)}\right\rfloor,$$
where $p_i$ is the $i$th prime, $\#$ is the primorial operator, and $\varphi$ is totient.
Example
$$f(3)=\left\lfloor\frac{5\#}{\...
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Why is this alternating summation involving reciprocals of divisors always positive? A conjecture.
Conjecture. For all $n,m \in \Bbb{N}$,
$$
f(n, m) := \sum_{c\mid d\mid n\# \\ \gcd(c, 2m) = 1}\dfrac{(-1)^{\omega(d)}}{d}
$$
is greater than $0$.
Example verification code:
...
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Is this old news? $\sum_{i=1}^n \frac{i!}{p_i\#} \approx 1.240053652689\dots$
This is a soft question as it arises out of my curiosity alone. I noticed that as $n$ increases, $\frac{n!}{p_n\#}$ decreases in magnitude much faster than $\frac{1}{p_n}$, and I wondered if the sum $$...
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Related to Problem 4697 from Crux Mathematicorum
Let $f_a:\mathbb N\to \mathbb N$ with $f_a(1) = 1$, $f_a(2) = a$ for some $a\in \mathbb N$ and, for each positive
integer $n\ge 3$, $f_a(n)$ is the smallest value not assumed at lower integers that ...
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Minimum value made of the reciprocals of the first $n$ primes
Let $n$ be a positive integer , $p_k$ the k-th prime number and $a_j=-1$ or $a_j=1$ for $j=1,\cdots ,n$
What is the minimum value of $$S:=|\sum_{j=1}^n \frac{a_j}{p_j}|$$ ?
Motivation : If we ...
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Why does the primorial $23\#$ come up so often in long prime arithmetic progressions?
This section of the Wikipedia article on the Green-Tao theorem gives examples of the longest known arithmetic progressions of prime numbers. For every known arithmetic progression of $24$ or more ...
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An open problem concerns primorials plus one having square factors. What about primorials squared plus one?
No primorial plus one has ever been found that has square factors. However, when I started looking at the square of primorials plus one, I found:
(23#)^2 + 1 = 29^2 * 53 * 1116604864937
I've only ...
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Combining disjoint congruences: $q\equiv \prod_ip_{u_i}^{r_{u_i}}\mod d$ and $q\equiv \prod_ip_{u_i}^{s_{u_i}}\mod p_{n+1}$
I have partitioned the prime factors of $p_n\#$ -- using the typical primorial definition: $x\#$ is the product of all primes not greater than $x$ -- into two sets identified by product: $d,\frac{p_n\#...
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Heuristics for Square-free Proof?
I like to waste my time on long-standing unsolved problems - the choice this week is the square-free nature of $1+p_n\#$, where $p_n\#$ is the $n$-th primorial.
The question here is short: does anyone ...
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Showing existence in congruence $q\equiv d^k+\frac{mp_n\#}d\pmod{p_n\#}$
The basic (updated) problem is
Given $n\in\Bbb N,q:(q,p_n\#)=1$, show that there exist $d\mid p_n\#,k\ge 1,m\in [1,p_n-1]$ such that $q\equiv d^k+\frac{mp_n\#}d\pmod{p_n\#}$.
(Note that $d$ is ...
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Fermat numbers are congruent to $17$ or $47$ modulo $210$. Can we generalize this result?
Let $n>1$ be an integer and $F_n:=2^{2^n}+1$.
Result: $n$ even: $F_n\equiv17\pmod{210}$
Result: $n$ odd: $F_n\equiv47\pmod{210}$
Proof: Suppose $n$ even, $F_n - 1\equiv16\pmod{210}$. Then $2^{2^{n+...
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A result about Fermat's numbers. Is my proof correct? Is that result useful ?Can we generalize that result?
Let $n$ be an integer and $F_n:=2^{2^n}+1$.
$$n=2,3,4:F_n\equiv17\pmod{30}$$
$\mathbf{Result:}\;n>1:F_n\equiv17\pmod{30}$
$\mathbf{Proof:}$ Suppose $F_n - 1\equiv16\pmod{30}$. Then $2^{2^{n+1}}=({2^...
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A divisibility rule for 7 in primoradic with an historical example. I'm looking for other examples of the same type.
I have given as much information as I could about primoradic here.
In primoradic , multiples of 7 which are not multiples of 2, 3 or 5 end
by (...:0:1:0:1), (...:1:3:0:1), (...:2:2:2:1), (...:3:0:0:1),...
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What is the primorial development of e ?(the notion of "primorial development" is defined in the body) [closed]
I am interested in Primorial number system (primoradic, see stub OEIS). In that system, you can define "primorial fractions" as fractions which can be writen as $$
\frac{{a_1 }}{2} + \frac{{...
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I'm looking for a proof for $\frac 14=\sum_{k=2}^{\infty}\frac{\left(\frac{p_k-1}{2}\right)}{P(p_k)}$
I'm interested in Primorial numeral system (primoradic, see stub OEIS). In that system, you can define "primorial fractions" as fractions which can be writen as $$
\frac{{a_1 }}{2} + \frac{{...
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I'm trying to generalize some simple results about $ 2 ^ n $. It's useful to write them in primoradic (see stub OEIS).
$$ 2 ^ 3 = 8 = 2 + 1 \times 6 \equiv 2 \pmod 6 \text ; $$
$$ 2 ^ 5 = 32 = 2 + 1 \times 30 \equiv 2 \pmod {30 } \text ; $$
$$ 2 ^ { 13 } = 8192 = 2 + 39 \times 210 \equiv 2 \pmod { 210 } \text ; $$
$$ ...
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Reference request: The primorial
The primorial function is defined as
$$p_n\#=\prod_{k=1}^{n}p_k$$
where $p_k$ is the $k$th prime (the more general definition is $n\#=\prod_{k=1}^{\pi(n)}p_k$, where $\pi(\cdot)$ is the prime counting ...
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Contradiction between OEIS and factordb.com
OEIS A014545 says 1+13494## is a prime number but factordb.com says it is composite, where n## is the product of first n primes on factordb. Which is correct?
edit:
Sorry if my question is not ...
user854931
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How can I prove this inequality? $\frac{p_k}{p_{k+1} + 1} < \frac{\ln(\ln(Primorial(k)))}{\ln(\ln(Primorial(k+1)))}$
How can I prove the following inequality? Or what are some tips that would help me prove it myself?
$$\frac{p_k}{p_{k+1} + 1} < \frac{\ln(\ln(Primorial(k)))}{\ln(\ln(Primorial(k+1)))}$$ for $k &...
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Is $\ n=107\ $ the only positive integer with more than one representation $n = q$#$ / p$# $\ +\ p$#?
The primorial $\ p\ $# is defined as the product of the primes upto $\ p\ $ , $\ p$#$:=2\cdot 3\cdot 5\cdot 7\cdots p\ $.
Define $f(p,q)=\ q$#$/p$#$\ +\ p$# for prime numbers $\ p\le q\ $
Is $n=107$ ...
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Is there a name for this family of sequences?
The sequence ${\displaystyle{M_n:=2^{p_n}-1}}$, where ${\displaystyle{n\gt0}}$ and ${p_n}$ is the ${\displaystyle{n}^{th}}$ prime number, is commonly known as the Mersenne numbers (not to be confused ...
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