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

Scott Sitar, Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2; email: scottsitar@gmail.com

Abstract/Résumé:

We show that for any integer \(M > 1\), any integer \(k\), and any admissible angle \(\theta\), there are infinitely many \(\theta\)-congruent numbers which are congruent to \(k\) modulo \(M\). Our method is inspired by an argument used by Chahal for an analogous result on congruent numbers modulo \(8\). Since congruent numbers are \(\pi/2\)-congruent numbers, this also includes as a special case the parallel statement for congruent numbers, originally due to Bennett.

Soit \(M\) un entier tel que \(M > 1\), soit \(k\) un entier, et soit \(\theta\) un angle admissible, nous montrons qu’il y a une infinité de nombres \(\theta\)-congruents dans la classe de \(k\) modulo \(M\). Notre méthode est inspirée par cela de Chahal, où il a montré le résultat analogue pour les nombres congruents modulo \(8\). Car les nombres congruents sont aussi des nombres \(\pi/2\)-congruents, notre travail contient aussi le résultat analogue pour les nombres congruents, démontré initialement par Bennett.

Keywords:

AMS Subject Classification: Cubic and quartic equations 11D25

PDF(click to download): Elliptic Curves and Families of Congruent and $theta$-congruent Numbers

Patrick Ingram, Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T 1Z4; email: pingram@math.ubc.ca

Abstract/Résumé:

We call the integer \(N\) a \(k\)th-power numerical centre for \(n\) if \[1^k+2^k+\cdots+N^k = N^k+(N+1)^k+\cdots+n^k.\] We prove, using the explicit lower bounds on linear forms in elliptic logarithms, that there are no nontrivial fifth-power numerical centres for any \(n\), and demonstrate that there are only finitely many pairs \((N, n)\) satisfying the above for any given \(k>1\). The problem of finding \(k\)-th-power centres for \(k=1, 2, 3\) has been treated in .

On dit qu’un entier \(N\) est un centre numérique de puissance \(k\) pour \(n\) si \[1^k+2^k+\cdots+N^k=N^k+(N+1)^k+\cdots+n^k.\] En utilisant des minorations explicites de formes linéaires de logarithmes elliptiques, on démontre qu’il n’y a aucun centre numérique non trivial de puissance \(5\), et on montre qu’il y a qu’un nombre fini des paires \((N, n)\) qui satisfont l’équation précèdente pour \(k>1\). Le problème de trouver des centres de puissance \(k\) pour \(k=1, 2, 3\) est traité dans [7].

Keywords: house problem, linear forms in elliptic logarithms, numerical centres

AMS Subject Classification: Cubic and quartic equations 11D25

PDF(click to download): On $k$-th power numerical centres

P.G. Walsh

Abstract/Résumé:

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

Keywords:

AMS Subject Classification: Fibonacci and Lucas numbers and polynomials and generalizations, Cubic and quartic equations, Units and factorization 11B39, 11D25, 11R27

PDF(click to download): On subsums of units in cubic number fields and ternary recurrence sequences

P.G. Walsh

Abstract/Résumé:

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

Keywords:

AMS Subject Classification: Cubic and quartic equations, Linear forms in logarithms; Baker's method 11D25, 11J86

PDF(click to download): A note on Ljunggren's theorem about the Diophantine equation aX2 -bY4 =1