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We use the estimation of the number of integers $n$ such that $\lfloor n^c\rfloor$ belongs to an arithmetic progression to study the coprimality of integers in ${\mathbb{N}}^c={\left\{\lfloor n^c\rfloor \right\}}_{n\in \mathbb{N}}$, $c>1$, $c\notin \mathbb{N}$.

The functional equation $f(y-x)-g\left(xy\right)=h\left(1/x-1/y\right)$ is solved for general solution. The result is then applied to show that the three functional equations $f\left(xy\right)=f\left(x\right)+f\left(y\right)$, $f(y-x)-f\left(xy\right)=f(1/x-1/y)$ and $f(y-x)-f\left(x\right)-f\left(y\right)=f(1/x-1/y)$ are equivalent. Finally, twice differentiable solution functions of the functional equation $f(y-x)-g_1\left(x\right)-g_2\left(y\right)=h\left(1/x-1/y\right)$ are determined.