# Are there positive integral solutions for this simple system?

What are solutions (if any) of the following system for $$n_i,m_i \in \mathbb{Z}_{\ge 0}$$ and $$i=1,2$$ ?

$$\left\{ \begin{array}{ll} m_1n_2 + m_2n_1 = n_1n_2 \\ n_i^2 \ge 1+m_jn_j \text{ for } i \neq j \\ n_i \text{ divides } 1+m_jn_j \text{ for } i \neq j \end{array} \right.$$

There are none. Well, except some trivial stuff.

Third condition essentially means that $$n_1$$ and $$n_2$$ are coprime (all right, it means more than that, but the rest we won't need).

Second condition is not needed at all.

First condition says:

$$m_1n_2 + \underbrace{m_2n_1 = n_1n_2}_{\text{divisible by }n_1}$$

which means $$m_1n_2$$ is also divisible by $$n_1$$, which means $$m_1$$ alone is, which in turn means that $$m_1\geqslant n_1$$ (unless $$m_1=0$$, which we'll discuss later). By similar logic, $$m_2\geqslant n_2$$, and so the LHS of the first equation hopelessly exceeds the RHS.

Now what if $$m_1=0$$? From the first equation it follows that $$m_2=n_2$$, and the question boils down to: $$n_1|n_2^2+1,\; n_2|1$$, which gives the only (up to permutation) solution: $$n_1=2,\;m_1=0,\;n_2=m_2=1$$.