Challenge Level

The following solution was recieved from Andrei of School 205 Bucharest. Well done and thank you Andrei.

10201 could be written (in base $x$) as:

$$\begin{align*}10201 &= 1x^0 + 2x^2 + 1x^4 \\ &= x^4 + 2x^2 + 1 \\ &= (x^2 + 1)^2 \end{align*}$$

Now, I write 10101 in a similar manner, in base $y$:

$$\begin{align*} 10101& = y^4 + y^2 + 1\\ & = y^4 + 2y^2 - y^2 + 1\\ & = (y^4 + 2y^2 + 1) - y^2\\ & = (y^2 + 1)^2 - y^2\\ & = (y^2 + 1 + y)(y^2 + 1 -y) \end{align*}$$

Therefore both expressions can be factorised, so they are composite.