# 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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### 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 ...
1 vote
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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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1 vote
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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),...
• 3,376
1 vote
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