What is the best way to denote a set of solutions in a professional setting?

Question

I have the following Diophantine equation that I want to solve over the positive (real) integers:

$$x^2+y^2=x+9y\tag1$$

Questions:

1. I want to solve this equation for real positive integers bigger than or equal to 2. How do I write that mathematically?
2. When I find the solutions I got: $(x,y)$ can be $(5,4)$, $(5,5)$ and $(1,9)$. How do I write that mathematically that those are the solutions?

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I think that the answer to question 1 is:

$$\left(x\in\mathbb{N}\space\wedge\space x\ge2\right)\space\wedge\space\left(y\in\mathbb{N}\space\wedge\space y\ge2\right)\tag2$$

I think that the answer to question 2 is of these three notations:

\begin{align} (x,y)&=(5,4),(5,5),(1,9),\tag3 \\ (x,y)&=\{(5,4),(5,5),(1,9)\},\tag4 \\ (x,y)&\in\{(5,4),(5,5),(1,9)\}.\tag5 \end{align}

Answer 1

It needs to be (5) because (3), (4) imply $(x,\,y)$ is equal to its set of possible values rather than an element of that set.

Answer 2

Let $A = \mathbb{N} \setminus{\{0\}}.$ You can write the set of solutions as

$$S = \{(x,y) \in A\times A \;|\; x^2+y^2=x+9y\}=\{(5,4),(5,5),(1,9)\}.$$

Notice that if you require $x \ge 2$, then you can't include the pair $(1,9)$.

Answer 3

I would probably write

1. $n,m\in\mathbb N:n,m\ge2$ (equivalently $\mathbb Z$),

2. $(n,m)=(5,4),(5,5),(1,9)$.

If you want a truly "professional" way, such as in a math review, you need to adhere to the conventions of that particular publication.