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{matrix} 10201 & = 1 x^{0} + 2 x^{2} + 1 x^{4} \\ = x^{4} + 2 x^{2} + 1 \\ = (x^{2} + 1)^{2} \end{matrix}

Now, I write 10101 in a similar manner, in base

y

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

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