IJPAM: Volume 107, No. 4 (2016)

ON THE DIOPHANTINE EQUATION $x^4+y^4=p^kz^7$

Keng Yarn Wong$^1$, Hailiza Kamarulhaili$^2$
$^{1,2}$School of Mathematical Sciences
Universiti Sains Malaysia
11800, Penang, MALAYSIA


Abstract. This paper solves the Diophantine equation $x^4+y^4=p^kz^7$ nontrivially in the case of $x=y$ where $p$ is prime and $k \in \mathbb{Z}^+$. The parametric solutions are formulated using number theory theorems, especially those concerning divisibility of integers, linear Diophantine equations, properties of prime numbers, and properties of congruence. There exist infinitely many nontrivial integral solutions to this Diophantine equation, where the parametric solutions found solve completely for different values of $p$ and $k$ in the case of $x=y$.

Received: February 23, 2016

AMS Subject Classification: 11D41

Key Words and Phrases: Diophantine equation, congruence, septic degree

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DOI: 10.12732/ijpam.v107i4.23 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2016
Volume: 107
Issue: 4
Pages: 1063 - 1072


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