On $(a,b)$-balancing numbers

Abstract

Year: 2010 Vol.: 77 Fasc.: 3-4 Title: On (a; b)-balancing numbers Author(s): TAunde Kov¶acs , K¶alm¶an Liptai and P¶eter Olajos A positive integer n is called a balancing number if 1+: : : +(n i 1) = (n + 1) + ¢ ¢ ¢ + (n + r) for some positive integer r. Balancing numbers and their generaliza- tions have been investigated by several authors, from many aspects. In this paper we introduce the concept of balancing numbers in arithmetic progressions, and prove several e®ective ¯niteness and explicit results about them. In the proofs of our results, among others, we combine Baker's method, the modular method developed by Wiles and others, a result of Bennett about the diophantine equation jaxn i bynj = 1, the Chabauty method and the theory of elliptic curves. Address: TAunde Kov¶acs Institute of Mathematics University of Debrecen H-4010 Debrecen, P.O. Box 12 Hungary Address: K¶alm¶an Liptai Institute of Mathematics and Informatics Eszterh¶azy K¶aroly College H-3300 Eger, Eszterh¶azy t¶er 1 Hungary Address: P¶eter Olajos Department of Applied Mathematics University of Miskolc H-3515 Miskolc-Egyetemv¶aros Hungary