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.
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