In this paper, we study the arithmetic partial differential equations x'_p = axⁿ 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 = axⁿ 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.