Geodesic
On some conjectures about arithmetic partial differential equations
Journal of integer sequences, Tome 20 (2017) no. 5.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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 $x'_{p} = a$.
Classification : 11A25, 11A51
Keywords: arithmetic derivative, partial derivative 
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Pandey, Ram Krishna; Saxena, Rohit. On some conjectures about arithmetic partial differential equations. Journal of integer sequences, Tome 20 (2017) no. 5. http://geodesic.mathdoc.fr/item/JIS_2017__20_5_a0/