Existence of Product of Three Distinct Primes between n and 2n

Theorem

Let $n \in \Z$ be an integer such that $n > 15$.

Then between $n$ and $2 n$ there exists at least one integer which is the product of $3$ distinct prime numbers.

Let $16 \le n \le 29$.

Then:

$n < 30 < 2 n$

and we have:

$30 = 2 \times 3 \times 5$

Hence the result holds for $n$ in that range.

Let $n \ge 30$.

$\exists q, r \in \N: n = 6 q + r$, $0 \le r < 6$, $q \ge 5$

By Bertrand-Chebyshev Theorem, there is a prime $p$ where $5 \le q < p < 2 q$.

Hence $p$ is not $2$ or $3$, and:

$\ds n$

$<$

$\ds 6 \paren {q + 1}$

$\ds $

$\le$

$\ds 2 \times 3 \times p$

$\ds $

$\le$

$\ds 12 q$

$\ds $

$\le$

$\ds 2 n$

This proves the result.

$\blacksquare$