Geodesic
On some conjectures about arithmetic partial differential equations
Journal of integer sequences, Tome 20 (2017) no. 5
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 
@article{JIS_2017__20_5_a0,
     author = {Pandey,  Ram Krishna and Saxena,  Rohit},
     title = {On some conjectures about arithmetic partial differential equations},
     journal = {Journal of integer sequences},
     year = {2017},
     volume = {20},
     number = {5},
     zbl = {1390.11014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2017__20_5_a0/}
}
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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/