Question 136
A positive integer $N$ is presented in the form $N = a_1 - a_2 = b_1 - b_2 = c_1 - c_2 = d_1 - d_2$, where $a_1$ and $a_2$ are perfect squares, $b_1$ and $b_2$ are perfect cubes, $c_1$ and $c_2$ are perfect 5th powers, and $d_1$ and $d_2$ are perfect 7th powers. Under the conditions above, determine whether two of the numbers $a_1, b_1, c_1, d_1$ should necessarily be equal.
(A. S. Golovanov)
(A. S. Golovanov)