Fourth Powers which are Sum of 4 Fourth Powers

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Theorem

The following positive integers are such that their fourth powers can be expressed as the sum of the fourth powers of $4$ other positive integers with no common factors:

$353, 651, 2487, 2501, 2829, \ldots$

This sequence is A039664 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof


Examples

353

$353^4 = 30^4 + 120^4 + 272^4 + 315^4$


651

$651^4 = 240^4 + 340^4 + 430^4 + 599^4$


Sources