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

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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:

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


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


Sources