Fourth Powers which are Sum of 4 Fourth Powers

From ProofWiki
Jump to navigation Jump to search

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