ORIGINAL ARTICLE Factoriangular numbers in balancing and Lucas-balancing sequence Sai Gopal Rayaguru 1 • Japhet Odjoumani 2 • Gopal Krishna Panda 1 Received: 9 April 2020 / Accepted: 4 July 2020 / Published online: 26 July 2020 Ó Sociedad Matemática Mexicana 2020 Abstract In this paper, we prove the nonexistence of factoriangular numbers in balancing and Lucas-balancing sequence. Keywords Balancing numbers  Lucas-balancing numbers  Factoriangular numbers  Linear forms in p-adic logarithms Mathematics Subject Classification 11B39  11D45 1 Introduction A balancing number n is a solution of the Diophantine equation: 1 þ 2 þþðn  1Þ¼ðn þ 1Þþðn þ 2Þþþðn þ rÞ with corresponding balancer r (see [1]). The balancing sequence fB n g n  1 satisfies the binary recurrence B nþ1 ¼ 6B n  B n1 ; n  1, with initial terms B 0 ¼ 0; B 1 ¼ 1. If B is a balancing number, then ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8B 2 þ 1 p is called a Lucas-balancing number [8]. The Lucas-balancing sequence fC n g n  1 also satisfies C nþ1 ¼ 6C n  C n1 ; n  1 & Sai Gopal Rayaguru saigopalrs@gmail.com Japhet Odjoumani japhet.odjoumani@imsp-uac.org Gopal Krishna Panda gkpanda_nit@rediffmail.com 1 Department of Mathematics, National Institute of Technology Rourkela, Rourkela, Odisha 769008, India 2 Institut de Mathe ´matiques et de Sciences Physiques, Universite ´ d’Abomey-Calavi, Dangbo, Benin 123 Boletín de la Sociedad Matemática Mexicana (2020) 26:865–878 https://doi.org/10.1007/s40590-020-00303-1