Let all functions be integer functions herein. I.e. $\Bbb{Z}\to\Bbb{Z}$ or $\Bbb{N}\to\Bbb{Z}$ where appropriate.
I found this jewel of floor functions.
So that made me wonder whether, we can solve for the simultaneous solution to:
$$ f(x) = g(y) $$
where
$$f(x) = x +2 \left\lfloor\frac{x}{3}\right\rfloor + 1 - [3 \mid x]\\ g(y) = y + 2 \left\lfloor\frac{y}{5}\right\rfloor + 1 - [5\mid y] $$
where $x_{k} = x \mod k$ the least non-negative residue. And $[3\mid x]=$ Iverson bracket for whether $3$ divides $x$.
But more specifically, can we describe a single-parameter function $h(t) = (x(t), y(t))$ such that:
$$ f(x(t)) = g(y(t)) $$
?
Attempt.
$$ x + 2\frac{ x - x_{3}}{3} + 1 - [3\mid x] = y + 2\frac{ y - y_{5}}{5} + 1 - [5\mid y] \\ \iff \\ 25x +10x_{3} - 15[3\mid x ] = 21y + 6y_{5} - 15[5\mid y] $$
Well, I've managed to reduce it for us in terms of variables modulo $3,5$, Iverson brackets and naked variables/coefficients, all "linearly" combined.
In other words how do we go about actually computing an equalizer (category theory term)?
Alternative Question. Clearly $h(t)$ is necessarily also injective and increasing as long as $x(t), y(t)$ are. Does this mean that in some way that the function $$k(z) = \# \{ 0 \leq y \leq z: y = f(x(t))[=g(y(t))] \text{ for some } t \in \Bbb{N}\} $$ has something to do with a possible parameterization?
