Someone on reddit asked how to show that $2^{15}-1$ was not a prime number, and I suddenly understood something I’d previously just ‘become used to’.

In binary, $2^{15}-1$ is 111,111,111,111,111. You can break that down in groups of three (or five) – but it’s fairly obviously $111 \times 1,001,001,001,001$.

Previously, I could certainly have proved that $a^{bc} - 1$ was a multiple of $a^b - 1$, probably using a geometric series formula, but I don’t think I had a visual feel that it was true – even if the geometric series is formally just the same as splitting the number up.