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I was thinking about Waring’s problem and Also about fractions.

So Naturally we can combine those.

Is Every positive fraction the Sum of at most 4 positive fractions squared ? How about cubes ? Etc

What is known about representations

$$ \frac{a}{b} = \sum_{i=1}^k (\frac{x_i}{y_i})^n $$

??

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Every positive integer is the sum of at most $g(k)$ $k$th powers of positive integers.

For a positive rational $\frac{m}{n}$ there is some integer $w$ such that $n | w^k.$ Indeed, for each prime $p$ dividing $n,$ with exponent $e,$ meaning $p^e | n$ but not $p^{e+1}|n,$ let the exponent $f$ of $p$ in $w$ be $f =\left\lceil \frac{e}{k} \right \rceil,$ so that $kf \geq e.$

Where was I, we have $n | w^k,$ so $$ M = w^k \cdot \frac{m}{n } = m \cdot \frac{w^k}{n} $$ is an integer. Write $M$ as the sum of $g(k)$ or fewer $k$th powers, $M = x_1^k + \cdots x_r^k$ in integers, $r \leq g(k). \;$ Then $$ \frac{m}{n} = \frac{M}{w^k} = \left( \frac{x_1}{w} \right)^k + \cdots + \left( \frac{x_r}{w} \right)^k $$

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  • $\begingroup$ Yes we can use this as an upper boundary. In fact a modified argument can easily show that G(K) is an upper bound , where G(K) means for sufficiently large n we need G(K) K th powers. However How about improving that ? The exact values ? Lower boundaries ? Etc. Thanks for the answer though. $\endgroup$ – mick Mar 24 '18 at 20:00

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