Question 19
Determine if there exist positive integers $n, a_1, a_2, \dots, a_{2012}$ all greater than 1, such that
$$
n^2 = a_1^2 + a_2^3 + a_3^5 + \dots + a_i^{p_i} + \dots + a_{2012}^{p_{2012}},
$$
where $p_i$ is the $i$-th prime.
$$
n^2 = a_1^2 + a_2^3 + a_3^5 + \dots + a_i^{p_i} + \dots + a_{2012}^{p_{2012}},
$$
where $p_i$ is the $i$-th prime.