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:

$\tuple {8373, 8433, 8493}, \tuple {8517, 8577, 8637}, \ldots$

Proof

 $\displaystyle 8373^4$ $=$ $\displaystyle 4450^4 + 5500^4 + 5670^4 + 7123^4$ $\displaystyle 8433^4$ $=$ $\displaystyle 4730^4 + 4806^4 + 5230^4 + 7565^4$ $\displaystyle 8493^4$ $=$ $\displaystyle 524^4 + 4910^4 + 5925^4 + 7630^4$

 $\displaystyle 8517^4$ $=$ $\displaystyle 1642^4 + 3440^4 + 6100^4 + 7815^4$ $\displaystyle 8577^4$ $=$ $\displaystyle 1050^4 + 2905^4 + 5236^4 + 8230^4$ $\displaystyle 8637^4$ $=$ $\displaystyle 3450^4 + 3695^4 + 5780^4 + 8012^4$

$\blacksquare$

The internal structure of these numbers reveals an interesting pattern:

 $\displaystyle 8373$ $=$ $\displaystyle 3 \times 2791$ $\displaystyle 8433$ $=$ $\displaystyle 3^2 \times 937$ $\displaystyle 8493$ $=$ $\displaystyle 3 \times 19 \times 149$

 $\displaystyle 8517$ $=$ $\displaystyle 3 \times 17 \times 169$ $\displaystyle 8577$ $=$ $\displaystyle 3^3 \times 953$ $\displaystyle 8637$ $=$ $\displaystyle 3 \times 2879$