3 Numbers in A.P. whose 4th Powers are Sum of Four 4th Powers

Theorem
The following triplets of integers in arithmetic sequence with common difference of $60$ can all be expressed as the sum of four $4$th powers:
 * $\tuple {8373, 8433, 8493}, \tuple {8517, 8577, 8637}, \ldots$

Proof
The internal structure of these numbers reveals an interesting pattern: