Abstract: The balancing numbers originally introduced by Behera and Panda ... numbers posses several fascinating properties, some are available in Panda [10].
PERIODICITY OF BALANCING NUMBERS G.K. Panda1 and S.S. Rout2
Mathematics Subject Classification: 11 A 05, 11 B 39, 11 B 50 Key words: Balancing numbers, Pell numbers, Associated Pell numbers, Periodicity, Modular arithmetic Abstract: The balancing numbers originally introduced by Behera and Panda in the year 1999 as solutions of a Diophantine equation on triangular numbers possess many interesting properties. Many of these properties are comparable to certain properties of Fibonacci numbers, while some others are more interesting. In the year 1960, Wall studied the periodicity of Fibonacci numbers modulo arbitrary natural numbers. The periodicity of balancing numbers modulo primes and modulo terms of certain sequences exhibits beautiful results, again, some of them are identical with corresponding results of Fibonacci numbers while some others are more interesting. An important observation concerning the periodicity of balancing numbers is that, the period of this sequence coincides with the modulus of congruence if the modulus is any power of 2. There are three known primes for which the periods of the sequence of balancing numbers modulo each prime is equal to the period modulo square of that prime, while for the Fibonacci sequence, till date no such prime is available.
1. INTRODUCTION The concept of balancing numbers came into existence after a paper by Behera and Panda [2] in the year 1999. They defined balancing numbers as solution of the Diophantine equation 1 2 ⋯ 1 1 2 ⋯ , calling the balancer corresponding to . Balancing numbers posses several fascinating properties, some are available in Panda [10] and some others in Panda and Ray [11]. The balancing number is denoted by and the balancing numbers satisfy a recurrence relation 6 , 0, 1. Panda and Rout [12] generalized this recurrence relation to , 0, 1 and proved that all properties of balancing numbers studied by Panda [10] are also true for this generalized sequence when 1. The theory of balancing numbers was extensively studied and generalized by many authors. The interested readers may refer to [3, 5, 6, 7, 8, 9, 14] for a detailed review. In the year 1960, while studying the periodicity of Fibonacci numbers, Wall [13] conjectured that there may be some prime such that the period of the Fibonacci sequence modulo might be equal to the period of the sequence modulo , although he could not find a counter example in the first 10,000 natural numbers. Recently, Elsenhans and Jahnel [4] extended this search in primes up to 10 but could not find any such prime. The balancing numbers, in this regard, behave quite well; the period of the balancing sequence modulo 13, 31 and 1546463 are same with the period of the sequence modulo 13 , 31 and 1546463 respectively. Another deficiency in the periodicity of Fibonacci numbers is that, no formula is available to calculate the period; but as we will see, the periods of the sequence of balancing 1
numbers (henceforth we will call balancing sequence) is computable for certain moduli. Many properties relating to the periodicity of balancing numbers are identical to those of Fibonacci numbers. Some important results and identities concerning balancing, Lucas-balancing, Pell and associated Pell numbers are needed in the subsequent sections. We prefer to state all these results in the following paragraph and use subsequently without further reference to this section. It is well known that if is a balancing number then 8 1 is a perfect square [2] and the positive square root of 8 1 is called a Lucas-balancing number [10]. The Lucas-
balancing number is given by !8 1 . The following identities and properties of balancing and Lucas-balancing numbers are available in [10]. If " and are natural numbers, then # 3 # , $# $ # $ , $ $ $ ,
and 1 . The identity % $ √8 $ ' $ √8 $ resembles DeMoivre’s theorem of complex numbers and we will call it the DeMoivre’s theorem balancing numbers. Another important property of balancing numbers is that $ divides if and only if " divides . The Binet form for balancing numbers is
*
%(√)' %(√)' √)
*
. Pell sequence is
defined recursively as + 1, + 2 and + 2+ + . Similarly the associated Pell sequence is defined by , 1, , 3 and , 2, , . The Binet forms for Pell and associated Pell numbers are respectively +
*
%√' %√' √
*
and ,
*
%√' %√'
*
and
some well- known identities like , + + , 2+ , + + + 1, , 2+ 1 are consequences of the Binet forms.
2. PERIODICITY OF BALANCING SEQUENCE In this section, we first show that the balancing sequence modulo any natural number is periodic and then study properties relating to the divisibility of the period. Throughout this paper, - denotes the least period (subsequently we will call period) of the balancing sequence modulo . It is well known that 0 and a natural question is: “Given any natural number ", is there any natural number such that ≡ 0 mod "? The following theorem answers this question in affirmative. 2.1 Theorem. For any natural number ", there exists a natural number n such that ≡ 0 mod ". Proof. It is well known that modulo ", there are " distinct least residues 0,1, ⋯ , " 1 and each balancing number is congruent to one of these residues. By pigeonhole principle, there exists two balancing numbers 2 and 3 4 5 both congruent to some 6, 0 7 6 7 " 1 2
modulo ". Hence 2 ≡ 3 mod ", and since for each , 8 1, it follows that 2 ≡ # 3 mod ". If 2 ≡ 3 mod ", then 23 2 3 2 3 ≡ 2 3 2 ≡ 0mod " and if 2 ≡ 3 , then 23 2 3 2 3 ≡ 2 3 2 ≡ 0mod ". This completes the proof. ∎ It is well known that for any integer 9, divides : . Thus, if ≡ 0 mod " then
: ≡ 0 mod ". By virtue of Theorem 2.1, the set S
