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

$10201\mathrm{could}\mathrm{be}\mathrm{written}(\mathrm{in}\mathrm{base}x)\mathrm{as}:$

${\displaystyle \begin{array}{cc}\multicolumn{1}{c}{10201}& =1x^0+2x^2+1x^4\\ \multicolumn{1}{c}{}& =x^4+2x^2+1\\ \multicolumn{1}{c}{}& =(x^2+1){}^2\end{array}}$

$\mathrm{Now},I\mathrm{write}10101\mathrm{in}a\mathrm{similar}\mathrm{manner},\mathrm{in}\mathrm{base}y:$

${\displaystyle \begin{array}{cc}\multicolumn{1}{c}{10101}& =y^4+y^2+1\\ \multicolumn{1}{c}{}& =y^4+2y^2-y^2+1\\ \multicolumn{1}{c}{}& =(y^4+2y^2+1)-y^2\\ \multicolumn{1}{c}{}& =(y^2+1){}^2-y^2\\ \multicolumn{1}{c}{}& =(y^2+1+y)(y^2+1-y)\end{array}}$

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