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 […]

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, […]

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 […]

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 $1+\prod_{k=1}^n p_k$ 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 $1+\prod_{k=1}^n p_k$?

In the classic proof of Bertrand’s postulate by Paul Erdős, he shows that $x\# < 4^x$ where $x\#$ is the primorial for $x$. Is there any tighter upper bound for a given primorial $x\#$? Ideally, does anyone know if there are any relatively recent papers on estimating the upper bound of a primorial?

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