S. Subburam, Department of Mathematics, SASTRA University, Thanjavur - 613401, Tamil Nadu, India; e-mail: ssubburam@maths.sastra.edu

R. Thangadurai, Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad - 211019, India; e-mail: thanga@hri.res.in

Abstract/Résumé:

In this paper, we shall prove that all positive integral solutions \((x, y, z)\) of the diophantine equation \(x^{3} + by + 1 – xyz = 0\) satisfy \(x \le b\left((2b^{3} + b)^{3} + 1\right) + 1,\) \(y \le (2b^{3} + b)^{3} + 1,\) and \(z \le \left(b\left((2b^{3} + b)^{3} + 1\right) + 1\right)^{2} + 2b^{3} + b\) for a given positive integer \(b\). As an application of this result, we investigate the divisors of the sequence \(\{n^3+1\}\) in residue classes. More precisely, we study the following sums: \[\displaystyle\sum_{b \le X }\displaystyle\sum_{\tiny\begin{array}{c}d \mid n^{3} + 1 \\ d \equiv – b \pmod{n} \end{array}} 1 \hspace{0.1in}\text{and} \hspace{0.1in} \displaystyle\sum_{n \le X}\displaystyle\sum_{\tiny\begin{array}{c}d \mid n^{3} + 1 \\ d \equiv – b \pmod{n} \end{array}} 1\] for a given positive real number \(X\) and a positive integer \(b\).

Keywords: Diophantine equations, divisors, residue classes

AMS Subject Classification: Cubic and quartic equations 11D25

PDF(click to download): On the Diophantine Equation $x^{3} + by + 1 - xyz = 0$

R.A. Mollin

Abstract/Résumé:

No abstract available but the full text pdf may be downloaded at the title link below.

Keywords: Diophantine equations, continued fractions, ideals

AMS Subject Classification: Factorization; primality, Continued fractions, Quadratic and bilinear equations, Quadratic extensions 11A51, 11A55, 11D09, 11R11

PDF(click to download): Proof of Some Conjectures by Kaplansky