Balancing numbers which are concatenation of two repdigits
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ORIGINAL ARTICLE
Balancing numbers which are concatenation of two repdigits Sai Gopal Rayaguru1 • Gopal Krishna Panda1 Received: 3 February 2020 / Accepted: 24 April 2020 Ó Sociedad Matemática Mexicana 2020
Abstract In this paper, we show that 35 is the only balancing number which is concatenation of two repdigits. Keywords Balancing numbers Repdigits Linear forms in logarithms
Mathematics Subject Classification 11J86 11B39 11B50 11D72
1 Introduction Balancing numbers n, are solutions of the Diophantine equation 1 þ 2 þ þ ðn 1Þ ¼ ðn þ 1Þ þ ðn þ 2Þ þ þ ðn þ rÞ for some natural number r, called as the balancer corresponding to n (see [2]). The sequence of balancing numbers is denoted by fBn gn 1 and they satisfy the binary recurrence Bnþ1 ¼ 6Bn Bn1 , n 1 with initial terms B0 ¼ 0 and B1 ¼ 1. If B is a balancing number, then 8B2 þ 1 is always a perfect square and its positive square pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi root C ¼ 8B2 þ 1 is called a Lucas-balancing number. The Lucas-balancing numbers satisfy the same recurrence as that of balancing numbers with initial terms C0 ¼ 1 and C1 ¼ 3. The Binet formulas for these sequences are given by
& Sai Gopal Rayaguru [email protected] Gopal Krishna Panda [email protected] 1
Department of Mathematics, National Institute of Technology Rourkela, Rourkela, Odisha 769008, India
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S. G. Rayaguru, G. K. Panda
a n bn a n þ bn pffiffiffi ; Cn ¼ ; ð1Þ 2 4 2 pffiffiffi pffiffiffi where a ¼ 3 þ 8 and b ¼ 3 8 are roots of the characteristic equation x2 6x þ 1 ¼ 0. These sequences can be extended to negative indices n as Bn ¼ Bn and Cn ¼ Cn respectively. Let g 2 be any positive integer. A natural number N is called a base g repdigit if all of its base g-digits are equal, that is, N of the form m g 1 N¼a ; for some m 1; where a 2 f1; 2; . . .; g 1g: g1 Bn ¼
When g ¼ 10, N is simply called a repdigit. Given positive integers A1 ; . . .; At , the concatenation of their base g strings of digits is A1 AtðgÞ . When g ¼ 10, the base g is omitted. Thus, a repdigit N is just a ðgÞ ; N ¼ a|fflffl{zfflffl} m times
whereas concatenation of two repdigits in base 10 is a ab b;
where a; b 2 f1; . . .; 9g:
For k 1 and g 2, a positive integer N of the form N ¼ a1 a1 a2 a2 ak ak ðgÞ ; |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} m1 times
m2 times
mk times
where a1 ; . . .; ak 2 f0; 1; . . .; g 1g; a1 6¼ 0 can be viewed as concatenation of k repdigits in base g. In recent years, there are several papers concerning Diophantine equations involving repdigits and the terms of binary recurrence sequences such as Fibonacci, Lucas, Pell, Pell–Lucas, balancing and Lucas-balancing sequence (see [5–9, 13, 14]). In [1], Alahmadi et al. showed that the only Fibonacci numbers which are concatenation of two repdigits are f13; 21; 34; 55; 89; 144; 233; 377g. In this paper, we implement the method used in [1] to prove the following result. Theorem 1 The only balancing number which is concatenation of two repdigits is 35.
2 Preliminaries
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