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

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

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