Certain Diophantine equations involving balancing and Lucas-balancing numbers

Transcription

1 ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 0, Number, December 016 Available online at Certain Diophantine equations involving balancing and Lucas-balancing numbers Prasanta Kumar Ray Abstract. It is well known that if x is a balancing number, then the positive square root of 8x + 1 is a Lucas-balancing number. Thus, the totality of balancing number x and Lucas-balancing number y are seen to be the positive integral solutions of the Diophantine equation 8x + 1 = y. In this article, we consider some Diophantine equations involving balancing and Lucas-balancing numbers and study their solutions. 1. Introduction The concept of balancing numbers came into existence after the paper [] by Behera and Panda wherein, they defined a balancing number n as a solution of the Diophantine equation 1++ +(n 1) = (n+1)+(n+)+ + (n + r), calling r the balancers corresponding to n. They also proved that, x is a balancing number if and only if 8x + 1 is a perfect square. In a subsequent paper [7], Panda studied several fascinating properties of balancing numbers calling the positive square root of 8x + 1, a Lucas-balancing number for each balancing number x. In [7], Panda observed that the Lucasbalancing numbers are associated with balancing numbers in the way Lucas numbers are attached to Fibonacci numbers. Thus, all balancing numbers x and corresponding Lucas-balancing numbers y are positive integer solutions of the Diophantine equation 8x + 1 = y. Though the relationship between balancing and Lucas-balancing numbers is non-linear, like Fibonacci and Lucas numbers, they share the same linear recurrence x n+1 = 6x n x n 1, while initial values of balancing numbers are x 0 = 0, x 1 = 1 and for Lucasbalancing numbers x 0 = 1, x 1 = 3. Demirtürk and Keskin [3] studied certain Diophantine equations relating to Fibonacci and Lucas numbers. Recently, Received June 18, Mathematics Subject Classification. 11B39; 11B83. Key words and phrases. Balancing numbers; Lucas-balancing numbers; Diophantine equations

2 166 PRASANTA KUMAR RAY Keskin and Karaatli [4] have developed some interesting properties for balancing numbers and square triangular numbers. Alvarado et al. [1], Liptai [5, 6], and Szalay [0] studied certain Diophantine equations relating to balancing numbers. The objective of this paper is to study some Diophantine equations involving balancing and Lucas-balancing numbers. The solutions are obtained in terms of these numbers.. Preliminaries In this section, we present some definitions and identities on balancing and Lucas-balancing numbers which we need in the sequel. As usual, we denote the n th balancing and Lucas-balancing numbers by and C n, respectively. It is well known from [7] that C n = 8B n + 1. The sequences { } and {C n } satisfy the recurrence relations +1 = 6 1, B 0 = 0, B 1 = 1; C n+1 = 6C n C n 1, C 0 = 1, C 1 = 3. The balancing numbers and Lucas balancing numbers can also be defined for negative indices by modifying their recurrences as 1 = 6 +1 ; C n 1 = 6C n C n+1, and calculating backwards. It is easy to see that B 1 = 1. Because B 0 = 0, we can check easily that all negatively subscripted balancing numbers are negative and that B n =. By a similar argument, it is easy to verify that C n = C n. Binet s formulas for balancing and Lucas-balancing numbers are, respectively, = αn 1 αn αn and C n = αn where α 1 = 1 +, α = 1, which are units of the ring Z( ). The totality of units of Z( ) is given by U = {α n 1, α n, α n 1, α n } : n Z}. The set U can be partitioned into two subsets U 1 and U such that U 1 = {u U : uu = 1} and U = {u U : uu = 1}. Since α 1 = α and α 1 α = 1, it follows that U 1 = {α n 1, α n, α n 1, α n : n Z}, U = {α n+1 1, α n+1, α n+1 1, α n+1 : n Z}. We also write λ 1 = α 1 = 3 + 8, λ = α = 3 8 and therefore, we have λ 1 λ = 1. Thus, the set U 1 can be written as U 1 = {λ n 1, λ n, λ n 1, λ n : n Z}.,

3 DIOPHANTINE EQUATIONS INVOLVING BALANCING NUMBERS 167 We also need the following identities (see [9]) while establishing certain identities and solving some Diophantine equations in the subsequent section. The first identity is B n = Using the recurrence relation +1 = 6 1, the identity reduces to B n B n 1 = 1, which we may call as Cassini s formula for balancing numbers. Similar identities for Lucas-balancing numbers are C n = C n 1 C n+1 8, C n 6C n C n+1 + C n 1 = 8. The idea of naming the identity as Cassini s formula comes from the literature, where this formula for Fibonacci numbers F n 1 F n+1 F n = ( 1) n or, equivalently, F n F n F n+1 F n 1 = ( 1) n 1 is available. Some other important identities are found in [8, 9]: +1 1 = C n, C n+1 C n 1 = 16, B m+n = B m C n + C m, B m n = B m C n C m, C m+n = C m C n + 8B m, C m n = C m C n 8B m, C m+n = B m+1 C n C n 1 B m. 3. Some identities involving balancing and Lucas-balancing numbers There are several known identities involving balancing, cobalancing and Lucas-balancing numbers. The interested readers are referred to [7] [19]. In this section, we only present some new identities involving balancing and Lucas-balancing numbers. The following two theorems are about nonlinear identities on balancing and Lucas-balancing numbers. Theorem 3.1. For any three integers k, m and n, C m+n + 16B k n C m+n B m+k = 8B m+k + C k n. Proof. By virtue of the identities B m±n = B m C n ± C m and C m±n = C m C n ± 8B m, we obtain [ ] [ ] [ ] Cn 8 Cm Cm+n =. B k C k B m B m+k

4 168 PRASANTA KUMAR RAY Since C n B k 8 = C n k which never vanishes, we have C k [ ] Cm = B m [ Cn 8 B k This implies that ] 1 [ ] Cm+n = 1 C k B m+k C n k [ ] [ ] Ck 8 Cm+n. B k C n B m+k C m = C kc m+n 8 B m+k C n k Since C m 8B m = 1, we have and B m = C nb m+k 8B k C m+n C n k. [ ] Ck C m+n 8 B [ ] m+k Cn B m+k 8B k C m+n 8 = 1, C n k from which the identity follows. C n k Theorem 3.. If k, m and n are three integers such that k n, then C m+n + C m+k + 8B k n = C k nc m+n C m+k. Proof. By virtue of the identities B m±n = B m C n ± C m and C m±n = C m C n ± 8B m, we obtain [ ] [ ] [ ] Cn 8 Cm Cm+n =. C k 8B k B m B m+k Since C n 8 C k 8B k = 8 k, and because k n, this determinant is nonvanishing. Therefore, we have [ ] [ Cm Cn 8B = n B m C k which implies that ] 1 [ ] Cm+n = 1 8B k B m+k k [ 8Bk 8 C k ] [ ] Cm+n, C n B m+k C m = B kc m+n C m+k 8 k Since C m 8B m = 1, we have and B m = C nc m+k C k C m+n 8 k. [ ] Bk C m+n C [ ] m+k Cn C m+k C k C m+n 8 = 1, 8 k and the required identity follows. 8 k Using the matrix multiplication [ ] 1 [ ] [ ] Bn C n Cm Bm+n =, B k C k B m B m+k it is easy to prove the following theorem.

5 DIOPHANTINE EQUATIONS INVOLVING BALANCING NUMBERS 169 Theorem 3.3. If k, m and n are three integers such that k n, then B m+n + B m+k B k n = C k nb m+n B m+k. 4. Some Diophantine equations involving balancing and Lucas-balancing numbers The identities of Section 3 induce the following three Diophantine equations: x + 16 xy 8y = C n, (4.1) x C n xy + y + C n = 1, (4.) x C n xy + y = B n. (4.3) Before the study of these equations, we present the Diophantine equation x 6xy + y = 1 (4.4) resulting out of Casini s formula for balancing numbers. The following theorem shows that all solutions of (4.4) are consecutive pairs of balancing numbers only. Theorem 4.1. All solutions of the Diophantine equation (4.4) are consecutive pairs of balancing numbers only. Proof. After factorization, the Diophantine equation (4.4) takes the form (λ 1 x y)(λ x y) = 1, where λ 1 = 3 + 8, λ = 3 8. This suggests that (λ 1 x y) and (λ x y) are units of Z( ), conjugate to each other, and are members of U 1. Thus for some integer n, we have the following four cases. Case 1 : (λ 1 x y) = λ n 1 and (λ x y) = λ n, Case : (λ 1 x y) = λ n and (λ x y) = λ n 1, Case 3 : (λ 1 x y) = λ n 1 and (λ x y) = λ n, Case 4 : (λ 1 x y) = λ n and (λ x y) = λ n 1. Solving the equation for Case 1, using Cramer s rule, we find λn 1 λ n 1 x = λ 1 = λn 1 λn λ 1 λ n 1 λ λ n =, y = λ 1 λ 1 λ 1 = λn 1 1 λ n 1 = 1. λ 1 λ 1 λ λ

6 170 PRASANTA KUMAR RAY The solution in Case is λn λ n 1 1 λ 1 λ n λ x = λ λ n 1 1 = = B n, y = 1 λ 1 = +1 = B (n+1). 1 λ Finally, the solutions in Case 3 and Case 4 are, respectively, x = B n, y = B (n 1) and x =, y = +1. Thus, in all cases, the solutions of the Diophantine equation (4.4) are consecutive pairs of balancing numbers only. Using Theorem 4.1, we can find solutions of a Diophantine equation derived from the identity of Theorem 3.1. Theorem 4.. All solutions of the Diophantine equation (4.1) are (x, y) = (C m n, B m ), ( C m+n, B m ), ( C m n, B m ), (C m+n, B m ) (4.5) for m, n Z. Proof. The equation (4.1) can be rewritten as λ (x + 8 y) = C n(8y + 1), implying that 8y +1 is a perfect square. Hence, y is a balancing number. So, we may take y = ±B m (y = B m is equivalent to y = B m ), 8y + 1 = C m. When y = B m, we have x = 8B m ± C m C n or x = 8B m ± C m C n and therefore, x = ±C m C n 8B m, i.e. x = C m n or C m+n. When y = B m, we have x = C m n or C m+n. Thus, the totality of solutions of (4.1) is given by (4.5) for m, n Z. Theorem 4.3. All solutions of the Diophantine equation (4.) are given by (x, y) = ( C m n, C m ), ( C m+n, C m ), (C m+n, C m ), (C m n, C m ) (4.6) for m, n Z. Proof. The equation (4.) can be rewritten as (x + C n y) = (C n 1)(y 1) = 8B n(y 1), implying that 8(y 1) is a perfect square. Thus, y is odd and hence y 1 8 is a perfect square. Since y 1 and y+1 are consecutive integers, it follows that y 1 8 is a square triangular number. That is, it is the square of a balancing number, say y 1 = Bm 8

7 DIOPHANTINE EQUATIONS INVOLVING BALANCING NUMBERS 171 and hence y = 8Bm + 1 for some m, so that y = ±C m. Consequently, (x + C n y) = 8Bn(y 1) is equivalent to x + C m C n = ±8B m if y = C m, and x C m C n = ±8B m if y = C m. Thus, the totality of solutions of (4.) is given by (4.6) for m, n Z. In the following theorem, we consider the Diophantine equation (4.3) which may be considered as a generalization of the Diophantine equation discussed in Theorem 4.1. Theorem 4.4. All solutions of the Diophantine equation (4.3) are given by (x, y) = (B m n, B m ), ( B m+n, B m ), ( B m n, B m ), (C m+n, B m ) (4.7) for m, n Z. Proof. The equation (4.3) can be rewritten as (x + C n y) = (C n 1)(y 1) = B n(8y + 1), which suggests that 8y + 1 is a perfect square. Thus, y = ±B m (as usual y = B m is equivalent to y = B m ) and hence 8y + 1 = Cm. Now x can be obtained from x + C y = ±C m and therefore, the totality of solutions is given by (4.7) for m, n Z. In the remaining theorems, we present some Diophantine equations where the proofs require certain divisibility properties of balancing and Lucasbalancing numbers. Theorem 4.5. The solutions of the Diophantine equation are given by (x, y) = ( B(k+1)n x + C n xy + y = 1 (4.8), B ) nk, ( B(k 1)n Proof. The equation (4.8) can be rewritten as, B ) nk (x + C n y) = (C n 1)(y 1) = 8B ny + 1, (m, n Z). (4.9) which suggests that y is a balancing number. Letting y = B m, we have y = Bm, and since y is an integer, it follows that divides B m. Hence by Theorem.8 in [7] (see also [4]), n divides m. Thus, m = nk for some integer k and y = k. Further, (x + C n y) = 8B ny + 1 = 8B nk + 1 = C nk, and hence x + C n y = ±C nk. It follows that, x = C nk ± C nk = B (k+1)n, B (k 1)n.

8 17 PRASANTA KUMAR RAY Thus, the totality of solutions of (4.8) is given by (4.9). The following theorem that resembles Theorem 4. deals with a Diophantine equation whose proof requires conditions under which a Lucas-balancing numbers divides balancing and Lucas-balancing numbers. Theorem 4.6. The solutions of the Diophantine equation x + 16 xy 8y = 1 are given by ( C(k+1)n (x, y) =, B ) ( (k+1)n Ckn,, B ) (k+1)n (m, n Z). (4.10) C n C n C n Proof. The equation x + 16 xy 8y = 1 can be rewritten as (x + 8 y) = 8C ny + 1, implying that 8Cny + 1 is a perfect square and C n y is a balancing number, say C n y = B m, hence C n divides B m. It is easy to see that this is possible if and only if m is an even multiple of n, and hence m = kn for some integer k. Thus, y = B kn C n. Further, (x + 8 y) = 8B kn + 1 = C kn, and hence x = 8 y ± C kn = 8B kn ± C kn. C n Therefore, the totality of solutions is given by (4.10). Lastly, we present a theorem which is a variant of Theorems 3.1 and 4.5. Theorem 4.7. The solutions of the Diophantine equation are given by x C n xy + y = 8B n (4.11) (x, y) = (C m n, C m ), (C m+n, C m ), ( C m+n, C m ), ( C m n, C m ) (4.1) for m, n Z. Proof. The equation (4.11) can be rewritten as (x C n y) = 8Bn(y 1), which suggests that 8(y 1) is a perfect square. Hence y is odd and 8(y 1) = 64 1 y 1 y + 1. Since y 1 and y+1 are consecutive integers, it follows that 1 y 1 y+1 is a square triangular number and hence is equal to the square of a balancing number (see []), say, 1 y 1 y + 1 = Bm for some m and we have 8(y 1) = 64Bm. Thus, y = 8Bm + 1 = Cm implying that y = ±C m, and the equation (x C n y) = 8Bn(y 1) is

9 DIOPHANTINE EQUATIONS INVOLVING BALANCING NUMBERS 173 reduced to (x C n y) = 64BmB n. Therefore, x C n y = ±8B m and the solutions of the equation (4.11) are given by (4.1) for m, n Z. References [1] S. Alvarado, A. Dujella, and F. Luca, On a conjecture regarding balancing with powers of Fibonacci numbers, Integers 1 (01), [] A. Behera and G. K. Panda, On the square roots of triangular numbers, Fibonacci Quart. 37 (1999), [3] B. Demirtürk and R. Keskin, Integer solutions of some Diophantine equations via Fibonacci and Lucas numbers, J. Integer Seq. 1(009), Article , 14 pp. [4] R. Keskin and O. Karaatli, Some new properties of balancing numbers and square triangular numbers, J. Integer Seq. 15 (01) Article 1.1.4, 13 pp. [5] K. Liptai, Fibonacci balancing numbers, Fibonacci Quart. 4 (004), [6] K. Liptai, Lucas balancing numbers, Acta Math. Univ. Ostrav. 14 (006), [7] G. K. Panda, Some fascinating properties of balancing numbers, in: Proceeding of the Eleventh International Conference on Fibonacci Numbers and their Applications, Congr. Numer. 194 (009), [8] G. K. Panda and P. K. Ray, Cobalancing numbers and cobalancers, Int. J. Math. Math. Sci. 8 (005), [9] G. K. Panda and P. K. Ray, Some links of balancing and cobalancing numbers with Pell and associated Pell numbers, Bull. Inst. Math. Acad. Sin. (N.S.). 6 (011), [10] G. K. Panda and S. S. Rout, Gap balancing numbers, Fibonacci Quart. 51 (013), [11] B. K. Patel and P. K. Ray, The period, rank and order of the sequence of balancing numbers modulo m, Math. Rep. (Bucur.) 18 (016), [1] P. K. Ray, Application of Chybeshev polynomials in factorization of balancing and Lucas-balancing numbers, Bol. Soc. Parana. Mat. 30 (01), [13] P. K. Ray, Certain matrices associated with balancing and Lucas-balancing numbers, Matematika 8 (01), 15. [14] P. K. Ray, Curious congruences for balancing numbers, Int. J. Contemp. Math. Sci. 7 (01), [15] P. K. Ray, Factorization of negatively subscripted balancing and Lucas-balancing numbers, Bol. Soc. Parana. Mat. 31 (013), [16] P. K. Ray, New identitities for the common factors of balancing and Lucas-balancing numbers, Int. J. Pure Appl. Math. 85 (013), [17] P. K. Ray, Some congruences for balancing and Lucas-balancing numbers and their applications, Integers 14 (014), Paper No. A8, 8 pp. [18] P. K. Ray, Balancing and Lucas-balancing sums by matrix methods, Math. Rep. (Bucur.) 17 (016), [19] P. K. Ray and B. K. Patel,Uniform distribution of the sequence of balancing numbers modulo m, Unif. Distrib. Theory 11 (016), [0] L. Szalay, On the resolution of simultaneous Pell equations, Ann. Math. Inform. 34 (007), Veer Surendra Sai University of Technology, Odisha, Burla , India address: rayprasanta008@gmail.com