Question 159
Find the smallest prime $q$ such that
$$
q = a_{1}^{2} + b_{1}^{2} = a_{2}^{2} + 2 b_{2}^{2} = a_{3}^{2} + 3 b_{3}^{2} = \ldots = a_{10}^{2} + 10 b_{10}^{2}
$$
where $a_{i}, b_{i}$ $(i=1,2, \ldots, 10)$ are positive integers.
$$
q = a_{1}^{2} + b_{1}^{2} = a_{2}^{2} + 2 b_{2}^{2} = a_{3}^{2} + 3 b_{3}^{2} = \ldots = a_{10}^{2} + 10 b_{10}^{2}
$$
where $a_{i}, b_{i}$ $(i=1,2, \ldots, 10)$ are positive integers.