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Number Theory Level 3

$x ! + n \leq p \leq x! +x$ Is there any prime number $p$ such that given inequality above can be true, where $1\leq n\leq x$ and $x>2$ ?

Inspiration

No Yes

1 solution

Jacopo Piccione
Aug 15, 2018

Obviously, there isn't any prime number $p$ such that $x!+2 \leq p \leq x!+x$ (for $x>2$ ) because $m|m \wedge m|x! \Rightarrow m|x!+m \; \forall m \in \{2,...,x\}$ . But $x!+1$ itself can be prime: in fact, if it's prime it's called a factorial prime .

Hence, answer is $\boxed {YES}$

Ah! They are hidden. Really?

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