linear forms in elliptic logarithms — 1 articles found.

On $k$-th power numerical centres

C. R. Math. Rep. Acad. Sci. Canada Vol. 27, (4), 2005 pp. 105–110
Vol.27 (4) 2005
Patrick Ingram Details
(Received: 2005-07-08 , Revised: 2005-09-15 )
(Received: 2005-07-08 , Revised: 2005-09-15 )

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