Diophantine equations — 2 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
S. Subburam; R. Thangadurai Details
(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

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


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 $x^{3} + by + 1 - xyz = 0$

Proof of Some Conjectures by Kaplansky

C. R. Math. Rep. Acad. Sci. Canada Vol. 23 (2) 2001, pp. 60–64
Vol.23 (2) 2001
R.A. Mollin Details
(Received: 2000-10-27 )
(Received: 2000-10-27 )

R.A. Mollin


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

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