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

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



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 $$ \frac{m}{n} = \frac{M}{w^k} = \left( \frac{x_1}{w} \right)^k + \cdots + \left( \frac{x_r}{w} \right)^k $$

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.