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Some Congruences Involvıng Catalan, Pell and Fibonacci Numbers

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Some Congruences Involvıng Catalan, Pell and Fibonacci Numbers MATHEMATICA MONTISNIGRI Vol XLVIII (2020) SOME CONGRUENCES INVOLVING CATALAN, PELL AND FIBONACCI NUMBERS S. KOPARAL1* AND N. ÖMÜR1 1 Kocaeli University Mathematics Department 41380 İzmit Kocaeli Turkey. *Corresponding author. E-mail: [email protected] DOI: 10.20948/mathmontis-2020-48-2 Summary. In this paper, using some special numbers and combinatorial identities, we show some interesting congruences: for a prime p5 , (p-1)/2 C P 22p+2 k 7-p /2 p , k 2 -4 +8- (mod p) k=0 8 k+1 p p p (p+1)/2 C 5 k-1 , k -qp 2 + (mod p) k=1 4 k k+1 6 (p+1)/2 C L 2p-2 5 5 91 k-1 3p - + - (mod p) k p+2 , k=1 -16 k k+1 k+2 2 p p 2 p 40 where Cn , Ln and Pn are the n th Catalan number, the th Lucas number and the th Pell . number, respectively. denotes the Legendre symbol. p 1 INTRODUCTION The Fibonacci sequence Fn and the Lucas sequence Ln are defined by the following recursions: Fn+1 F n +F n - 1 and Ln+1 L n +L n - 1, n 0 , where F00 , F11 and L20 , L11 , respectively. If α and β are the roots of equation x2 - x -1 0 , the Binet formulas of the sequences and have the forms αnn - β F and L αnn +β , n α - β n respectively. The Pell sequence Pn and the Pell-Lucas sequence Qn are defined recursively by Pn+1 2P+P n n - 1 and Qn+1 =2Q n +Q n - 1 , , in which P00 , P11 and Q01 Q 2 , respectively. If γ and δ are the roots of equation 2010 Mathematics Subject Classification: 11B39, 05A10, 05A19. Key words and Phrases: Central binomial coefficients, congruences, Fibonacci numbers, Pell numbers. 10 S. Koparal and N. Ömür x2 - 2x-1 0 , the Binet formulas of the sequences and have the forms γnn -δ P and Q γnn +δ , n γ -δ n respectively. The harmonic numbers have interesting applications in many fields of mathematics, such as number theory, combinatorics, analysis and computer science. Harmonic numbers H n are defined as for a positive integer n n 1 Hn , k=1 k 3 11 25 where H0 . The first few harmonic numbers are 1, , , ,... 0 2 6 12 Some elementary combinatorial properties of the Catalan numbers are given in [2, 6, 8]. The Catalan numbers are given by 1 2n 2n 2n Cn - , n N. n+1n n n+1 For a prime p and an integer a with pa | , we write the Fermat quotient p-1 a qp a a - 1 /p. For an odd prime and an integer , denotes the Legendre p symbol defined by 0 if p a, a 1 ifa isa quadraticresiduemodp , p 1 ifa isa quadraticnonresiduemodp . In [3], E. Lehmer showed that for prime p 3, Hp - 1 /2 - 2qp 2 mod p . (1.1) In [9], Z.W. Sun obtained that p - 1 /2 1 2k m m - 4 , k mod p k=0 mpk and p - 1 /2 12k m m - 4 m m - 4 , k - mod p k=0 m k+1 k 2 2 p where is an odd prime and m is any integer not divisible by p. Let be a fixed prime 3 . Define P p p Q p - 1 k n x - x - 1 - 1 n x qx and Gx , n n p k=1 k where x is variable. In [1], A. Granville showed that 11 S. Koparal and N. Ömür q x - G1 x mod p , p G2 x G2 1- x +x G2 1 - 1/x mod p , 2 pp q x -2x G22 x -2 1 - x G 1 - x mod p . It was proved in [1] (the last expression on page 3) that for any integer n1 , k k n n n - 1 n - 1 - x 1 n - 1 1 - x -1 1 - x - - x -1 + . 221.2 k=1 k - 1 k nk=1 k n In [7], Z.H. Sun obtained the following congruences: for an odd prime and Gnp x Z x , pkp5 1 1+ x - 1 - xp p - 1 1 - x -1 p - 1 xr r - 1 1 G x - +p mod p2 , 2 2 p p k=1 kr=2 r s=1 s and for a prime p3 and nN , n p2 npGn+1 x -1 x Gn 1/x - Gn x mod p . In [10], Z.W. Sun showed that for a prime , p a p - 1 HL p - 1 HFF2 p - 1 kk 0 mod p and k k k mod p . k=1 k k=1 k pk=1 k In [5], H. Pan and Z.W. Sun obtained that for a prime , p - 1 L k 0 mod p , k 2 n k=1 and for a prime p 2,5 , pa - 1 2k a k p 3 -1 1-2Fa mod p , a p k=0 k 5 p - 5 where is a positive integer. In [4], S. Mattarei and R. Tauraso showed that for a prime , p - 12k xk 2G λ +G μ mod p , 2 22 k=1 k k 1 1 where λ= 1+ 1 - 4x and μ= 1- 1 - 4x . 2 2 2 SOME CONGRUENCES INVOLVING SPECIAL NUMBERS In this section, we will give the congruences involving some special numbers. Now, we give the following lemma for further use. Lemma 1. Let be any positive integer. Then k+1 k+1 n+1 n n - 1 -x 1xn - 1 1 - x -1 1 - x -1 22= + + Hn k=1 k - 1 k k+1 nk=1 k k+1 n n+1 n 12 S. Koparal and N. Ömür and k+2 k+2 n+2 n n - 1 -x 1 n - 1 1 - x -1 1 - x -1 x x2 22= + + - Hn . k=1 k - 1 k k+1 k+2 nk=1 k k+1 k+2 n n+1 n+2 n+1 2n Proof. From Binomial Theorem, we have k+1 n+1 n - 1 1 - x -1 1 - x -1 + +xHn k=1 k k+1 n n+1 k+1 k+1 n 1 - x -1 n1 n 1 - x -1+ k+1 x =+x = k=1 k k+1 k=1 k k=1 k k+1 kj n+1 n+1 k n+1 k 1 - x -1+kx 11k j -x k-1 == -x = k=2 k k -1 k=2 k k -1 j=2 j k=2k -1 j=2 j j-1 j j n+1 k -x k-2 n+1 -x n1 k-2 = = . k=2 j=2 j j-1 j-2 j=2 j j-1 kj j-2 p a n+1 k-2 n From the equality = , we get k=j j-2 j-1 k+1 n+1 n - 1 1 - x -1 1 - x -1 + +xHn k=1 k k+1n n n+1 j j+1 j+1 n+1 n -x n n -x n n-1 -x = = =n . 2 j=2 j-1 j j-1 j=1 j j j+1 j=1 j-1 j j+1 Thus, the first identity is obtained. Similarly, we obtain other identity. a For a prime and a rational number x= written in reduced form, x 0 mod p if and b only if is divisible by . 1 1 λ= 1+ 1 - 4x μ= 1- 1 - 4x Theorem2 1. Let be an odd prime.2 For any rational number such that x- 1 0 mod p , p - 1 /2 p-1 C p+1 2 k xk+1 1-λ - μ + λp +μ p - 1 mod p , (2.1) k=0 k+1 p where and . Proof.x Replacing and by (p+1)/2 and 4x in (1.2), respectively, we have 13 S. Koparal and N. Ömür k+1 k p+1 /2 p+1 p - 1 /2 p - 1 /2 - 4x p - 1 /2 1 - 4x 1 - 4x -1 2 = +2 - Hp - 1 /2 . 2 k=0 k k+1 k=1 k p+1 p - 1 /2 k 2k From the congruences -4 mod p for k=1,…, p - 1 /2 , k k 11 mod p for k=1,…,p - 1 and (1.1), we write p+k k p - 1 /2 k+1 p - 1 /2 k 2k x p+1 /2 1 - 4x -2 2 1 - 4x - 1 + +2q 2 mod p . (2.2) 2 p k=0 k k+1 k=1 k p - 1 k By congruence -1 mod p for , we get k kk p - 1 /2 1 - 4x p - 1 /2 p - 1 1 - 4x - (2.3) k=1 kkk=1 2k - 1 pp p - 1 /2 2 - 1+ 1 - 4x - 1 - 1 - 4x p2 p k - 1 - 4x = mod p . ppk=1 2k Substituting (2.3) into (2.2), we have p - 1 /2 p - 1 /2 k+1 Ck k+1 2k x x= 2 k=0 k+1 k=0 k k+1 pp 1 - 4x - 1 - 1 - 4x+1 +2 p+1 /2 1 - 1 - 4x - q 2 - p 2p p - 1 p+1 2 =1 - λ - μ + λpp +μ - 1 mod p . p Thus, this concludes the proof. From Theorem 1, we immediately deduce the following results. Corollary 1. Let be an odd prime. Then p - 1 /2 Ck k 4 1 - qp 2 mod p , k=0 4 k+1 p - 1 /2 k -1 Ck p - 1 1 - Lp 5 2 +5 - 1 mod p , (2.4) k=0 k+1 p p p - 1 /2 p+2 Ck 7 - p /2P p 22 k 2 - 4 +8 - mod p , k=0 8 k+1 p p p 14 S.
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