Let $k$ be an odd positive integer. Can any positive rational number $n$ be written as $\frac{a^k+b^k}{c^k+d^k}$ where $a,b,c,d$ are positive rational numbers/ rational numbers?

The answer is true for $k=1$, and probably also true for $k=3$ by explicit construction (see http://mathworld.wolfram.com/DiophantineEquation3rdPowers.html). How about the case $k=5$ and $k=7$? For general $k$?

Motivation: By general philosophy of arithemetic for surfaces, there will be little rational points on a surface of general type. And for $k=3$ and fixed rational number $n$, we are considering rational points on cubic surfaces which are del Pezzo so shall contain rational curves.

Let $k$ be an odd positive integer. Can every positive rational number $n$ be written as $\frac{a^k+b^k}{c^k+d^k}$ where $a,b,c,d$ are positive rational numbers/ rational numbers?

The answer is true for $k=1$, and probably also true for $k=3$ by explicit construction (see http://mathworld.wolfram.com/DiophantineEquation3rdPowers.html). How about the case $k=5$ and $k=7$? For general $k$?

Motivation: By general philosophy of arithemetic for surfaces, there will be little rational points on a surface of general type. And for $k=3$ and fixed rational number $n$, we are considering rational points on cubic surfaces which are del Pezzo so shall contain rational curves.

Let $k$ be a odd positive integer. Can any positive rational number $n$ be written as $\frac{a^k+b^k}{c^k+d^k}$ where $a,b,c,d$ are positive rational numbers/ rational numbers?

The answer is true for $k=1$, and probably also true $k=3$ by explicit construction (see http://mathworld.wolfram.com/DiophantineEquation3rdPowers.html). How about the case $k=5$ and $k=7$? For general $k$?

Motivation: By general philosophy of arithemetic for surfaces, there will be little rational points on a surface of general type. And for $k=3$ and fixed rational number $n$, we are considering rational points on cubic surfaces which are del Pezzo so shall contain rational curves.

Let $k$ be an odd positive integer. Can any positive rational number $n$ be written as $\frac{a^k+b^k}{c^k+d^k}$ where $a,b,c,d$ are positive rational numbers/ rational numbers?

The answer is true for $k=1$, and probably also true for $k=3$ by explicit construction (see http://mathworld.wolfram.com/DiophantineEquation3rdPowers.html). How about the case $k=5$ and $k=7$? For general $k$?

Motivation: By general philosophy of arithemetic for surfaces, there will be little rational points on a surface of general type. And for $k=3$ and fixed rational number $n$, we are considering rational points on cubic surfaces which are del Pezzo so shall contain rational curves.