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11D25, Cubic and quartic equations — 5 results found.

      
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On the Diophantine Equation $x^{3} + by + 1 – xyz = 0$
C. R. Math. Rep. Acad. Sci. Canada Vol. 36 (1) 2014, pp. 15–19
S. Subburam; R. Thangadurai (Received: 2013/02/21, Revised: 2014/02/18)

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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 \(b\).

Elliptic Curves and Families of Congruent and $\theta$-congruent Numbers
C. R. Math. Rep. Acad. Sci. Canada Vol. 32 (2) 2010, pp. 33–39
Scott Sitar (Received: 2009/06/22)

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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.

On $k$-th power numerical centres
C. R. Math. Rep. Acad. Sci. Canada Vol. 27, (4), 2005 pp. 105–110
Patrick Ingram (Received: 2005/07/08, Revised: 2005/09/15)

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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].

On subsums of units in cubic number fields and ternary recurrence sequences
C. R. Math. Rep. Acad. Sci. Canada Vol. 25 (1) 2003, pp. 13–18
P.G. Walsh (Received: 2002/07/15)

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No abstract available but the full text is online at the title link above.

A note on Ljunggren’s theorem about the Diophantine equation aX2-bY4 =1
C. R. Math. Rep. Acad. Sci. Canada Vol. 20 (4) 1998, pp. 113–118
P.G. Walsh (Received: 1998/02/20)

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No abstract available but the full text is online at the title link above.

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algebraic number theory approximation property automorphisms Bessel functions Boson-fermion correspondence C*-algebra Carmichael number center problem Chebyshev transform classification Classification of simple C*-algebras composition operators continued fractions Cuntz Semigroup elliptic curves fixed point Fourier transform function fields. functoriality general relativity generic property ideals indefinite inner product inductive limits of sub-homogeneous C*- algebras Irrational rotation algebra J-Hermitian matrix K-theory Kahler manifolds L-functions maximal ideal space nonexpansive mapping numerical range orthogonal polynomials Predual space prime number property SP Renormalization rotation algebras Salem number semi-reciprocal polynomials tracially approximate splitting interval algebras unbounded traces uniqueness Weak Markov set Whitney problems

Most used AMS

05C05 11A07 11A55 11B37 11B68 11D09 11D25 11D41 11E04 11F11 11F66 11F67 11G05 11R09 11R11 13B25 14J26 14M25 14P10 17B37 17B67 19K14 19K56 26A51 30C15 30H05 35B 37E10 37E20 37F25 39B72 42C05 43A07 46B20 46L05 46L35 46L40 46L55 46L80 47H10 53B25 53C55 54C60 60F10 83C05

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