Articles of primorial

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 consequence of some tests regarding the kind answer of @Will Jagy to a previous question at MSE. $E1$ can […]

Trying to understand why a set of residues modulo a primorial $p_k\#$ has a range of values smaller than $2p_{k+1}$

I’ve been reviewing the following: $$v_i = \left\lfloor\frac{ip_k\#}{p_{k+1}}\right\rfloor + c_i$$ where: $c_i \in \left\{1,2\right\}$ so that $v_i$ is odd and $v_ip_{k+1} > ip_k\# > (v_i-c)p_{k+1}$ $i$ is an integer such that $1 \le i \le p_{k+1}-1$ $\left\lfloor\frac{a}{b}\right\rfloor$ is a floor function I hit a result that surprises me. Let $[v_i]$ be a residue modulo $p_k\#$ […]

Question about the reduced residue system for a given primorial

It is well known that the number of elements in the reduced residue system for a given primorial $p_k\#$ is divisible by $p_k – 1$. Does it follow that if you divide the elements of a reduced residue class into distinct classes modulo $p_k$, that each class modulo $p_k$ has the same number of element […]

What is the product of $p_i-1 \over p_i$

This question already has an answer here: Convergence of infinite product of prime reciprocals? 1 answer

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 $b$. Let $f(m,p_n) =$ the number of integers $x$ where $0 < x \le m$ and […]

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 struggling to complete the argument. This is easy to show if $0 \le w […]

Is my conjecture true? : Every primorial is a superior highly regular number, and every superior highly regular number is a primorial.

I have invented two sets of positive integers: highly regular numbers and superior highly regular numbers. A positive integer $m \leq n$ is a regular of the positive integer $n$ if all prime numbers that divide $m$ also divide $n$. So for example: 4 has 3 regulars: 1, 2, 4. 6 has 5 regulars: 1, […]

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

Observe this list: $$ \begin{aligned} 2+1&=3\\ 2\cdot3+1&=7\\ 2\cdot3\cdot5+1&=31\\ 2\cdot3\cdot5\cdot7+1&=211\\ 2\cdot3\cdot5\cdot7\cdot11+1&=2311\\ 2\cdot3\cdot5\cdot7\cdot11\cdot13+1&=59\cdot509\\ 2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17+1&=19\cdot97\cdot277\\ 2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19+1&=347\cdot27953\\ 2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23+1&=317\cdot703763\\ 2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot29+1&=331\cdot571\cdot34231\\ 2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot29\cdot31+1&=200560490131\\ 2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot29\cdot31\cdot37+1&=181\cdot60611\cdot676421\\ 2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot29\cdot31\cdot37\cdot41+1&=61\cdot450451\cdot11072701\\ 2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot29\cdot31\cdot37\cdot41\cdot43+1&=167\cdot78339888213593 \end{aligned} $$ Is it true that all prime factors occur with multiplicity one in this list? (Note that if one multiplies consecutive primes not starting from 2 and adds 1, there are many examples of multiplicities […]

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 $n!$ (in other words, the first primorial greater than the given factorial $n!$ such as $\not\exists \ p_j\#: \ n! \lt […]

How often is $1+\prod_{k=1}^n p_k$ not prime?

How often is $1+\prod_{k=1}^n p_k$ is not prime?,wkere $p_k$ is k’th prime consider that $2\times3\times5\times7\times11\times13+1=59\times509 = 30031$ is this one off or, are there infinitely many composites of the form $1+\prod_{k=1}^n p_k$?