# Fourth Powers which are Sum of 4 Fourth Powers/Examples/353

## Examples of Fourth Powers which are Sum of 4 Fourth Powers

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

## Proof

 $\displaystyle 30^4 + 120^4 + 272^4 + 315^4$ $=$ $\displaystyle 810 \, 000$ $\displaystyle$  $\, \displaystyle + \,$ $\displaystyle 207 \, 360 \, 000$ $\displaystyle$  $\, \displaystyle + \,$ $\displaystyle 5 \, 473 \, 632 \, 256$ $\displaystyle$  $\, \displaystyle + \,$ $\displaystyle 9 \, 845 \, 600 \, 625$ $\displaystyle$ $=$ $\displaystyle 15 \, 527 \, 402 \, 881$ $\displaystyle$ $=$ $\displaystyle 353^4$

Now we have that:

 $\displaystyle 442^2 - 272^2$ $=$ $\displaystyle 170 \times 714$ $\displaystyle$ $=$ $\displaystyle 17^2 \times 420$

Hence:

 $\displaystyle 442^2 - 3 \times 17^2$ $=$ $\displaystyle 272^2 + 289 \times 417$ $\displaystyle$ $=$ $\displaystyle 272^2 + 353^2 - 64^2$

But:

$3 \times 17 = 2 \times 26 - 1$

So:

 $\displaystyle 442^2 - 2 \times 26 \times 17 + 17$ $=$ $\displaystyle 442^2 - 2 \times 442 + 17$ $\displaystyle$ $=$ $\displaystyle 441^2 + 4^2$ $\displaystyle$ $=$ $\displaystyle 21^4 + 2^4$ $\displaystyle$ $=$ $\displaystyle 272^2 + 353^2 - 8^4$

Hence:

$353^2 + 272^2 = 2^4 + 8^4 + 21^4$

but:

 $\displaystyle 353^2 - 272^2$ $=$ $\displaystyle 81 \times 625$ $\displaystyle$ $=$ $\displaystyle 15^4$

So:

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

$\blacksquare$

## Historical Note

This result was discovered by Robert Norrie, who reported on it in $1911$.