559 is not Expressible as Sum of Fewer than 19 Fourth Powers

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Theorem

$559$ cannot be expressed as the sum of fewer than $19$ fourth powers:

$559 = 15 \times 1^4 + 2 \times 2^4 + 2 \times 4^4$

or:

$559 = 9 \times 1^4 + 4 \times 2^4 + 6 \times 3^4$


Proof

First note that $5^4 = 625 > 559$.

Then note that $3 \times 4^4 = 768 > 559$.

Hence any expression of $559$ as fourth powers uses no $n^4$ for $n \ge 5$, and uses not more than $2$ instances of $4^4$.

For the remainder, using $2^4$ uses fewer fourth powers than $16$ instances of $1^4$ does.


Now we have:

\(\displaystyle 559\) \(=\) \(\displaystyle 2 \times 4^4 + 0 \times 3^4 + 2 \times 2^4 + 15 \times 1^4\)
\(\displaystyle \) \(=\) \(\displaystyle 1 \times 4^4 + 3 \times 3^4 + 3 \times 2^4 + 12 \times 1^4\)
\(\displaystyle \) \(=\) \(\displaystyle 1 \times 4^4 + 2 \times 3^4 + 8 \times 2^4 + 13 \times 1^4\)
\(\displaystyle \) \(=\) \(\displaystyle 1 \times 4^4 + 1 \times 3^4 + 13 \times 2^4 + 14 \times 1^4\)
\(\displaystyle \) \(=\) \(\displaystyle 1 \times 4^4 + 0 \times 3^4 + 18 \times 2^4 + 15 \times 1^4\)
\(\displaystyle \) \(=\) \(\displaystyle 6 \times 3^4 + 4 \times 2^4 + 9 \times 1^4\)
\(\displaystyle \) \(=\) \(\displaystyle 5 \times 3^4 + 9 \times 2^4 + 10 \times 1^4\)
\(\displaystyle \) \(=\) \(\displaystyle 4 \times 3^4 + 14 \times 2^4 + 11 \times 1^4\)
\(\displaystyle \) \(=\) \(\displaystyle 3 \times 3^4 + 19 \times 2^4 + 12 \times 1^4\)
\(\displaystyle \) \(=\) \(\displaystyle 2 \times 3^4 + 24 \times 2^4 + 13 \times 1^4\)
\(\displaystyle \) \(=\) \(\displaystyle 1 \times 3^4 + 29 \times 2^4 + 14 \times 1^4\)
\(\displaystyle \) \(=\) \(\displaystyle 0 \times 3^4 + 34 \times 2^4 + 15 \times 1^4\)

and it can be seen that the first and the sixth use the least number of fourth powers, at $19$.

$\blacksquare$


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