Question 120
a) Let $a$ be a positive integer. Prove that none of the numbers $a^2 + 1, a^2 + 2, \dots, a^2 + 2a$ is a square.
b) Are there positive integers $m, n, p$ such that
$$
m^2 + n + p, \ n^2 + p + m, \ p^2 + m + n
$$
are all squares?
b) Are there positive integers $m, n, p$ such that
$$
m^2 + n + p, \ n^2 + p + m, \ p^2 + m + n
$$
are all squares?