IJPAM: Volume 80, No. 4 (2012)

$(py + g(z))(qy + h(z)) = ax^{2} + bx + c$

R. Srikanth$^1$, S. Subburam$^2$
$^{1,2}$Department of Mathematics
SASTRA University
Thanjavur, 613 401, Tamil Nadu, INDIA

Abstract. Let $a$, $b$, $c$, $p$, $q$ be given positive integers with $a = pq$ and let $g(z)$ and $h(z)$ be positive integer valued arithmetical functions such that there is no integer between $g(z)/p$ and $h(z)/q$ for any positive integer $z$. Consider the ternary diophantine equation $(py + g(z))(qy + h(z)) = ax^{2} + bx + c$ in variables $x$, $y$, $z$. In this paper, we find a real number $\gamma$ such that all positive integral solutions $(x, y, z)$ of the equation satisfy $x \le \gamma$.

Received: May 15, 2012

AMS Subject Classification: 11D61, 11D41

Key Words and Phrases: Diophantine equation, arithmetical function

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 80
Issue: 4