Are $8, 36, 216,$ and $2304$ the only perfect powers form of $p\#$$\pm6$, where $p\#$ is primorial?
I noticed that when I try to add or subtract $6$ from a primorial, the results are:
- $2+6=8$, which is a perfect cube.
- $30+6=36$, a perfect square.
- $210+6=216$, a perfect cube also.
- $2310-6=2304$, a perfect square number.
Note that I excluded trival examples.
My question is: Is these numbers are the only perfect powers that form of $p\#$$\pm6$, where $p\#$ is a primorial? If so, it can be a perfect higher odd prime perfect power?
My knowledge is that $p\#$$\pm6$ is only a perfect square if $p\#$$\pm6$ is equal to $6\pmod{8}$, and $p\#$ must not be divisible by $13$, so that means there is no perfect square above $30030$ that is $6$ more or less than a primorial, but I don’t know if there is argument that can be used for cubes and higher odd powers.