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**
On Some Conjectures about Arithmetic Partial Differential Equations
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Ram Krishna Pandey and Rohit Saxena

Department of Mathematics

Indian Institute of Technology Roorkee

Roorkee - 247667

India

**Abstract:**

In this paper, we study the arithmetic partial differential equations
*x'*_{p} = *a**x*^{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} =
*a**x*^{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*.

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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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