Factoriangular numbers in balancing and Lucas-balancing sequence
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ORIGINAL ARTICLE
Factoriangular numbers in balancing and Lucas-balancing sequence Sai Gopal Rayaguru1
•
Japhet Odjoumani2 • Gopal Krishna Panda1
Received: 9 April 2020 / Accepted: 4 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 þ ðn 1Þ ¼ ðn þ 1Þ þ ðn þ 2Þ þ þ ðn þ rÞ with corresponding balancer r (see [1]). The balancing sequence fBn gn 1 satisfies the binary recurrence Bnþ1 ¼ 6Bn Bn1 ; n 1, with initial terms B0 ¼ 0; B1 ¼ 1. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi If B is a balancing number, then 8B2 þ 1 is called a Lucas-balancing number [8]. The Lucas-balancing sequence fCn gn 1 also satisfies Cnþ1 ¼ 6Cn Cn1 ; n 1
& Sai Gopal Rayaguru [email protected] Japhet Odjoumani [email protected] Gopal Krishna Panda [email protected] 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
S. G. Rayaguru et al.
with initial terms C0 ¼ 1; C1 ¼ 3. Furthermore, the Binet forms of balancing and Lucas-balancing sequence are given by: a m bm a m þ bm pffiffiffi ; Cm ¼ ; m 1; ð1Þ 2 4 2 pffiffiffi pffiffiffi where a ¼ 3 þ 8 and b ¼ 3 8. Numbers of the form Ftn ¼ n! þ nðnþ1Þ are called factoriangular numbers (see 2 [3]). Go´mez Ruiz and Luca [4] proved that 2, 5, and 34 are the only Fibonacci numbers which are factoriangular. Subsequently, Luca et al. [6] proved that 2, 5, and 12 are the only Pell numbers which are factoriangular. In [5], Kafle et al. showed that the only Lucas numbers which are factoriangular are 1 and 2. In this paper, we use the method similar to that of [4] to prove that there does not exist any factoriangular number in both balancing and Lucas-balancing sequence, except the trivial solution B1 ¼ Ft0 and C0 ¼ Ft0 . Bm ¼
2 Preliminaries An upper bound for a nonzero p-adic linear form in logarithms of algebraic numbers due to Bugeaud and Laurent [2] serves as a main tool for the proof of our main results. The logarithmic height hðgÞ of an algebraic number g of degree d over Q with minimal primitive polynomial over the integers: f ðxÞ ¼ a0
d Y ðX gðiÞ Þ 2 Z; i¼1
where the leading coefficient a0 is positive and gðiÞ ; i ¼ 1; 2; . . .; d are conjugates of g is given by: ! d X 1 log a0 þ hðgÞ ¼ maxf1; logjgðiÞ jg : d i¼1 Put K ¼ gb11 gb22 ; where b1 ; b2 are positive integers and g1 ; g2 2 K n f0; 1g with K as the algebraic number field of degree dK . Let p be the prime ideal of the ring OK of algebraic integers in K, and for g 2 K, ordp ðgÞ denotes the order at which p appears in the prime factorization of the principal fractional ideal gOK generated by g in K. When g is
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