# Consecutive Integers whose Product is Primorial

## Theorem

The following primorials can be expressed as the product of consecutive integers:

$2, 6, 30, 210, 510 \, 510$

No others are known.

The corresponding indices of those primorials are:

$2, 3, 5, 7, 17$

The corresponding values of $n$ such that $p\# = \paren {n - 1} n$ are:

$2, 3, 6, 15, 715$

## Proof

 $\displaystyle 2 \#$ $=$ $\displaystyle 1 \times 2$ $\displaystyle$ $=$ $\displaystyle 2$ $\displaystyle 3 \#$ $=$ $\displaystyle 2 \times 3$ $\displaystyle$ $=$ $\displaystyle 6$ $\displaystyle 5 \#$ $=$ $\displaystyle 2 \times 3 \times 5$ $\displaystyle$ $=$ $\displaystyle 5 \times \paren {2 \times 3}$ $\displaystyle$ $=$ $\displaystyle 5 \times 6$ $\displaystyle$ $=$ $\displaystyle 30$ $\displaystyle 7 \#$ $=$ $\displaystyle 2 \times 3 \times 5 \times 7$ $\displaystyle$ $=$ $\displaystyle \paren {2 \times 7} \times \paren {3 \times 5}$ $\displaystyle$ $=$ $\displaystyle 14 \times 15$ $\displaystyle$ $=$ $\displaystyle 210$ $\displaystyle 17 \#$ $=$ $\displaystyle 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17$ $\displaystyle$ $=$ $\displaystyle \paren {2 \times 3 \times 7 \times 17} \times \paren {5 \times 11 \times 13}$ $\displaystyle$ $=$ $\displaystyle 714 \times 715$ $\displaystyle$ $=$ $\displaystyle 510 \, 510$

$\blacksquare$