Journal of Integer Sequences, Vol. 20 (2017), Article 17.5.2

On Some Conjectures about Arithmetic Partial Differential Equations

Ram Krishna Pandey and Rohit Saxena
Department of Mathematics
Indian Institute of Technology Roorkee
Roorkee - 247667


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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(Concerned with sequences A000040 A003415.)

Received January 31 2017; revised version received February 23 2017. Published in Journal of Integer Sequences, March 26 2017.

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