Dipper Operation/Examples/(m, n) = (3, 4)/Examples/x +(3,4) x = x

Examples of Equations on Big Dipper Operation $+_{3, 4}$
Let $\N_{<7}$ denote the initial segment of the natural numbers:
 * $\N_{<7} := \set {0, 1, \ldots, 6}$

Let $+_{3, 4}$ be the big dipper operation on $\N_{<7}$ defined as:
 * $\forall a, b \in \N_{<7}: a +_{3, 4} b = \begin{cases}

a + b & : a + b < 3 \\ a + b - 4 k & : a + b \ge 3 \end{cases}$

where $k$ is the largest integer satisfying:
 * $3 + 4 k \le a + b$

Consider the equation:
 * $x +_{3, 4} x = x$

This has the solutions:
 * $x = 0$
 * $x = 4$

Proof
Apparent from direct inspection of the Cayley table:

The main diagonal contains the elements $x$ of $\N_{<7}$ such that $x +_{3, 4} x$.

As can be seen, $x +_{3, 4} x = x$ when $x = 0$ and $x = 4$.