There are also patterns that suggest structure. For example $111 .\ldots111111$ in binary sometimes has a factor and sometimes does not, depending on the length of the line of ones. $1111$ is the product of $11$ and $101$ but $11111$ has no factors.
Teachers may wish to offer students, as a conjecture to explore, the suggestion that $2^{ab} - 1$ has $2^a - 1$ as a factor.
For example : $2^6 - 1$ is $63$ and its factors include $2^2 - 1$ ($3$) and $2^3 - 1$ ($7$)