(a+b)/c$ and $f := 4acd - n = (na+c)/b$, then $e,f$ are natural numbers, and we obtain the other identities - by routine algebra. By construction we have $\pi^{{\operatorname{I}}}_n(a,b,c,d,e,f) = (x,y,z)$, and the claim follows.

In particular, for fixed $n$, a Type I solution exists if and only if there is an ${\mathbb{N}}$-point $(a,b,c,d,e,f)$ of $\Sigma^{{\operatorname{I}}}_n$ with $abcd$ coprime to $n$ (the requirement that $a,b,c$ have no common factor can be removed using the symmetry ). By parameterising $\Sigma^{{\operatorname{I}}}_n$ using three or four of the six coordinates, we recover some of the known characterisations of Type I solvability:

Let $n$ be a natural number. Then the following are equivalent:

-   There exists a Type I solution $(x,y,z)$.

-   There exists $a,b,e \in {\mathbb{N}}$ with $e \mid a+b$ and $4ab \mid ne+1$. [@aigner]

-   There exists $a,b,c,d \in {\mathbb{N}}$ such that $4abcd = na+nb+c$ with $c$ coprime to $n$. [@bernstein]

-   There exist $a,c,d,e \in {\mathbb{N}}$ such that $ne+1=4ad (ce-a)$ with $c$ coprime to $n$. [@rosati; @mordell]

-   There exist $a,c,d,f \in {\mathbb{N}}$ such that $n=4acd-f$ and $f \mid 4a^2d+1$, with $c$ coprime to $n$. [@nakayama]

-   There exist $b,c,d,e$ with $ne = (4bcde-1)- 4b^2 d$ and $c$ coprime to $n$. 