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

Examples of Dipper Operation

Let $m, n \in \Z$ be integers such that $m \ge 0, n > 0$.

Let $\N_{< \paren {m \mathop + n} }$ denote the initial segment of the natural numbers:

$\N_{< \paren {m \mathop + n} } := \set {0, 1, \ldots, m + n - 1}$

Let $+_{m, n}$ be the dipper operation on $\N_{< \paren {m \mathop + n} }$ defined as:

$\forall a, b \in \N_{< \paren {m \mathop + n} }: a +_{m, n} b = \begin{cases}

a + b & : a + b < m \\ a + b - k n & : a + b \ge m \end{cases}$

where $k$ is the largest integer satisfying:

$m + k n \le a + b$

Let $m = 3$ and $n = 4$.

The Cayley table for $+_{3, 4}$ can be presented as follows:

$\begin{array}{r|rrrrrrr}

+_m & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 0 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ 1 & 1 & 2 & 3 & 4 & 5 & 6 & 3 \\ 2 & 2 & 3 & 4 & 5 & 6 & 3 & 4 \\ 3 & 3 & 4 & 5 & 6 & 3 & 4 & 5 \\ 4 & 4 & 5 & 6 & 3 & 4 & 5 & 6 \\ 5 & 5 & 6 & 3 & 4 & 5 & 6 & 3 \\ 6 & 6 & 3 & 4 & 5 & 6 & 3 & 4 \\ \end{array}$

Examples

Example: $x +_{3, 4} 2 = 3$

Consider the equation:

$x +_{3, 4} 2 = 3$

This has the solutions:

$x = 1$

$x = 5$

Example: $x +_{3, 4} x = 4$

Consider the equation:

$x +_{3, 4} x = 4$

This has the solutions:

$x = 2$

$x = 4$

$x = 6$

Example: $x +_{3, 4} x = x$

Consider the equation:

$x +_{3, 4} x = x$

This has the solutions:

$x = 0$

$x = 4$