# Mathematical ReportsComptes rendus mathématiques

### residue classes — 1 articles found.

On the Diophantine Equation $x^{3} + by + 1 – xyz = 0$

C. R. Math. Rep. Acad. Sci. Canada Vol. 36 (1) 2014, pp. 15–19
Vol.36 (1) 2014
(Received: 2013-02-21 , Revised: 2014-02-18 )
(Received: 2013-02-21 , Revised: 2014-02-18 )

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

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 $$b$$.
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