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Qiaochu Yuan
Qiaochu Yuan
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This is not an answer but a series of comments.

Here's an obstacle stopping you from straightforwardly generalizing the classical based on the Gaussian integers to, say, sums of two cubes. The problem is that $x^3 + y^3$ is not the norm of an element of a number field. If it were, the number field would be $\mathbb{Q}(\omega)$$\mathbb{Q}(\zeta_6)$, where $\omega$$\zeta_6$ is a primitive thirdsixth root of unity, but this is a quadratic number field and the norm of, say, $x + y \omega$$x + y \zeta_6$ is actually the quadratic factor $x^2 - xy + y^2$ of $x^3 + y^3$ rather than the whole thing.

The ring where norms give $x^3 + y^3$ is instead the ring $\mathbb{Q}[\omega]/(\omega^3 + 1)$$\mathbb{Q}[\zeta]/(\zeta^3 + 1)$, which unfortunately is not a field, so it's unclear in what sense the "algebraic integers" $\mathbb{Z}[\omega]/(\omega^3 + 1)$$\mathbb{Z}[\zeta]/(\zeta^3 + 1)$ inside it can have a reasonable theory of prime factorization. Moreover, the general element of this ring has the form $x + y \omega + z \omega^2$$x + y \zeta + z \zeta^2$, with three parameters, and the corresponding norm is

$$x^3 + y^3 + z^3 - 3xyz$$

so even if one has understood this norm one still has to take into account the additional constraint that $z = 0$.

It's also worth pointing out that $m = x^3 + y^3$ describes a curve of genus $1$ for almost all values of $m$ and IIRC it's already unknown whether there exists an algorithm taking as input such a curve and determining whether it has an integer point or not. Bjorn Poonen has written about related topics; see, for example, this expository article.

Some other relevant stuff:

This is not an answer but a series of comments.

Here's an obstacle stopping you from straightforwardly generalizing the classical based on the Gaussian integers to, say, sums of two cubes. The problem is that $x^3 + y^3$ is not the norm of an element of a number field. If it were, the number field would be $\mathbb{Q}(\omega)$, where $\omega$ is a primitive third root of unity, but this is a quadratic number field and the norm of, say, $x + y \omega$ is actually the quadratic factor $x^2 - xy + y^2$ of $x^3 + y^3$ rather than the whole thing.

The ring where norms give $x^3 + y^3$ is instead the ring $\mathbb{Q}[\omega]/(\omega^3 + 1)$, which unfortunately is not a field, so it's unclear in what sense the "algebraic integers" $\mathbb{Z}[\omega]/(\omega^3 + 1)$ inside it can have a reasonable theory of prime factorization. Moreover, the general element of this ring has the form $x + y \omega + z \omega^2$, with three parameters, and the corresponding norm is

$$x^3 + y^3 + z^3 - 3xyz$$

so even if one has understood this norm one still has to take into account the additional constraint that $z = 0$.

It's also worth pointing out that $m = x^3 + y^3$ describes a curve of genus $1$ for almost all values of $m$ and IIRC it's already unknown whether there exists an algorithm taking as input such a curve and determining whether it has an integer point or not. Bjorn Poonen has written about related topics; see, for example, this expository article.

Some other relevant stuff:

This is not an answer but a series of comments.

Here's an obstacle stopping you from straightforwardly generalizing the classical based on the Gaussian integers to, say, sums of two cubes. The problem is that $x^3 + y^3$ is not the norm of an element of a number field. If it were, the number field would be $\mathbb{Q}(\zeta_6)$, where $\zeta_6$ is a primitive sixth root of unity, but this is a quadratic number field and the norm of, say, $x + y \zeta_6$ is actually the quadratic factor $x^2 - xy + y^2$ of $x^3 + y^3$ rather than the whole thing.

The ring where norms give $x^3 + y^3$ is instead the ring $\mathbb{Q}[\zeta]/(\zeta^3 + 1)$, which unfortunately is not a field, so it's unclear in what sense the "algebraic integers" $\mathbb{Z}[\zeta]/(\zeta^3 + 1)$ inside it can have a reasonable theory of prime factorization. Moreover, the general element of this ring has the form $x + y \zeta + z \zeta^2$, with three parameters, and the corresponding norm is

$$x^3 + y^3 + z^3 - 3xyz$$

so even if one has understood this norm one still has to take into account the additional constraint that $z = 0$.

It's also worth pointing out that $m = x^3 + y^3$ describes a curve of genus $1$ for almost all values of $m$ and IIRC it's already unknown whether there exists an algorithm taking as input such a curve and determining whether it has an integer point or not. Bjorn Poonen has written about related topics; see, for example, this expository article.

Some other relevant stuff:

Source Link
Qiaochu Yuan
Qiaochu Yuan
  • 118.2k
  • 40
  • 447
  • 741

This is not an answer but a series of comments.

Here's an obstacle stopping you from straightforwardly generalizing the classical based on the Gaussian integers to, say, sums of two cubes. The problem is that $x^3 + y^3$ is not the norm of an element of a number field. If it were, the number field would be $\mathbb{Q}(\omega)$, where $\omega$ is a primitive third root of unity, but this is a quadratic number field and the norm of, say, $x + y \omega$ is actually the quadratic factor $x^2 - xy + y^2$ of $x^3 + y^3$ rather than the whole thing.

The ring where norms give $x^3 + y^3$ is instead the ring $\mathbb{Q}[\omega]/(\omega^3 + 1)$, which unfortunately is not a field, so it's unclear in what sense the "algebraic integers" $\mathbb{Z}[\omega]/(\omega^3 + 1)$ inside it can have a reasonable theory of prime factorization. Moreover, the general element of this ring has the form $x + y \omega + z \omega^2$, with three parameters, and the corresponding norm is

$$x^3 + y^3 + z^3 - 3xyz$$

so even if one has understood this norm one still has to take into account the additional constraint that $z = 0$.

It's also worth pointing out that $m = x^3 + y^3$ describes a curve of genus $1$ for almost all values of $m$ and IIRC it's already unknown whether there exists an algorithm taking as input such a curve and determining whether it has an integer point or not. Bjorn Poonen has written about related topics; see, for example, this expository article.

Some other relevant stuff: