2014/083) Show that no member of the infinite series: 10001, 100010001, 1000100010001, 10001000100010001, … is prime.

that is between each sucessive 1 there are 3 zeroes.

first we note that the smallest number 10001 = 73 * 137 and not a prime

then from the 2nd term onwards (now n >=2)
Write the series as
$\displaystyle\sum_{i=0}^{n}10^{4i}= \dfrac{10^{4n+4}-1}{10^4-1}$
=  $\dfrac{(10^{2n+2}+1)(10^{2n+2}-1)}{10^4-1}$
as $n \ge 3$ we have 2 factors $\ge 10^6-1$ and denominator 9999 so we shall have 2 distinct factors and hence composite