Summary: In this paper, we study the arithmetic partial differential equations $x'_{p} = ax^{n}$ and $x'_{p} = a$. We solve a conjecture of Haukkanen, Merikoski, and Tossavainen (HMT, in short) about the number of solutions (conjectured to be finite) of the equation $x'_{p} = ax^{n}$ and improve a theorem of HMT about finding the solutions of the same equation. Furthermore, we also improve another theorem of HMT about the solutions of the equation $x'_{p} = a$ and discuss one more conjecture of HMT about the number of solutions of $x'_{p} = a$.

11A25, 11A51

arithmetic derivative, partial derivative