NEW IDENTITIES FOR THE COMMON FACTORS OF BALANCING AND LUCASBALANCING NUMBERS


 John Wheeler
 11 months ago
 Views:
Transcription
1 International Journal of Pure and Applied Mathematics Volume 85 No , ISSN: (printed version); ISSN: (online version) url: doi: PAijpam.eu NEW IDENTITIES FOR THE COMMON FACTORS OF BALANCING AND LUCASBALANCING NUMBERS Prasanta Kumar Ray International Institute of Information Technology Gothapatna, PO: MALIPADA, Bhubaneswar, , INDIA Abstract: Balancing numbers n and balancers r are originally defined as the solution of the Diophantine equation (n 1) = (n + 1) + (n + )+ +(n+r). If n is a balancing number, then 8n +1 is a perfect square. Further, If n is a balancing number then the positive square root of 8n +1 is called a Lucasbalancing number. These numbers can be generated by the linear recurrences B n+1 = 6B n B n 1 and C n+1 = 6C n C n 1 where B n and C n are respectively denoted by the n th balancing number and n th Lucasbalancing number. In this study, we establish some new identities for the common factors of both balancing and Lucasbalancing numbers. AMS Subject Classification: 11B39, 11B83 Key Words: balancing numbers, Lucasbalancing numbers, recurrence relation 1. Introduction Behera and Panda [1] recently introduced a number sequence called balancing numbers defined in the following way: A positive integer n is called a balancing number with balancer r, if it is the solution of the Diophantine equation (n 1) = (n+1)+(n+)+...+(n+r). They also proved that the Received: January 11, 013 c 013 Academic Publications, Ltd. url:
2 488 P.K. Ray recurrence relation for balancing numbers is B n+1 = 6B n B n 1, n >, (1.1) where B n is the n th balancing number with B 1 = 1 and B = 6. It is well known that (see [1]), n is a balancing number if and only if n is a triangular number, that is 8n +1 is a perfect square. In [10], Lucasbalancing numbers are defined as follows: If n is a balancing number, C n = 8n +1 is called a Lucasbalancing number. The recurrence relation for Lucasbalancing numbers is same as that of balancing numbers, that is C n+1 = 6C n C n 1, n >, (1.) where C n is the n th Lucasbalancing number with C 1 = 3 and C = 17. Liptai [4], showed that the only balancing number in the sequence of Fibonacci numbers is 1. In [11] and [1], Ray obtain nice product formulas for both balancing and Lucasbalancing numbers. Panda and Ray [8], link balancing numbers with Pell and associated Pell numbers. They shown that balancing numbers are indeed the product of Pell and associated Pell numbers. Many interesting properties and important identities are available in the literature. Interested readers can follow [, 3, 5, 6, 7, 13, 14]. The closed form of both balancing and Lucasbalancing numbers are respectively given by and B n = λn 1 λn λ 1 λ (1.3) C n = λn 1 +λn (1.4) for n 1 with λ 1 = 3 + 8, λ = 3 8. These relations (1.3) and (1.4) are popularly known as Binet s formulas for balancing and Lucasbalancing numbers. In this paper, we obtain some new identities for the common factors of these numbers.. New Identities for the Common Factors of Balancing and LucasBalancing Numbers In this section, we present some new identities for the common factors of both balancing and Lucasbalancing numbers with the help of Binet s formula. It is clear that λ 1 +λ = 6, λ 1 λ = 8, λ 1 λ = 1. (.1)
3 NEW IDENTITIES FOR THE COMMON FACTORS OF Theorem.1. For n 1, the following identity is valid: Proof. By (.1), we obtain which finishes the proof. B 4n 6 = B n 1 C n+1. B n 1 C n+1 = λn 1 1 λ n 1 λ n+1 λ 1 λ = λ4n 1 λ4n λ 1 λ λ 1 λ λ 1 λ = B 4n 6 1 +λ n+1 Theorem.. For n 1, the following identity is valid: Proof. By (.1), we get B 4n+1 +1 = B n+1 C n. B n+1 C n = λn+1 1 λ n+1 λ n 1 +λ n λ 1 λ = λ4n+1 1 λ 4n+1 + (λ 1λ ) n λ 1 (λ 1 λ ) n λ λ 1 λ λ 1 λ = B 4n+1 +1 which is the end of the proof. Theorem.3. For n 1, the following identity is valid: Proof. By (.1), we have which is the end of the proof. B 4n+ +6 = B n+ C n. B n+ C n = λn+ 1 λ n+ λ n 1 +λ n λ 1 λ = λ4n+ 1 λ 4n+ +(λ 1 λ ) nλ 1 λ λ 1 λ λ 1 λ = B 4n+ +6
4 490 P.K. Ray By the same way, we have the following result. Theorem.4. For n 1, the following identity is valid: B 4n+3 1 = B n+1 C n+. The following lemma is already established in[8]. For the sake of simplicity we present the proof again. Lemma.5. For n 1, the following identity is valid: Proof. By (.1), we get which completes the proof. B n = B n C n. B n C n = λn 1 λn λ n 1 +λn λ 1 λ = λn 1 λn λ 1 λ = B n Lemma.6. For n 1, the following identity is valid: Proof. By (.1), we have which is the end of the proof. B 4n+1 1 = B n C n+1. B n C n+1 = λn 1 λn λ n+1 λ 1 λ 1 +λ n+1 = λ4n+1 1 λ 4n+1 (λ 1 λ ) nλ 1 λ λ 1 λ λ 1 λ = B 4n+1 1 By virtue of Lemma.5 and Lemma.6, we have the following result. Corollary.7. For n 1, we have B 4n+1 1 = B n C n C n+1.
5 NEW IDENTITIES FOR THE COMMON FACTORS OF Lemma.8. For n 1, the following identity is valid: C 4n+1 3 = 16B n B n+1. Proof. Since (λ 1 λ ) = 3, we get which ends the proof. 16B n B n+1 = 16 λn 1 λn λ 1 λ λ n+1 = λ4n+1 1 +λ 4n+1 = C 4n λ n+1 λ 1 λ (λ 1 λ ) nλ 1 +λ Since B n = B n C n, the following identity is valid for n 1: Theorem.9. C 4n+1 3 = C n C n+1. Theorem.10. For n 1, the following identity is valid: Proof. By (.1), we have which is the end of the proof. C 4n+1 +3 = C n C n+1. C n C n+1 = λn 1 +λn λ n+1 = λ4n+1 1 +λ 4n+1 = C 4n λ n+1 +(λ 1 λ ) nλ 1 +λ Lemma.11. For n 1, the following identity is valid: Proof. By using (.1), we have which completes the proof. B 4n+3 +1 = B n+ C n+1. B n+ C n+1 = λn+ 1 λ n+ λ n+1 λ 1 λ 1 +λ n+1 = λ4n+ 1 λ 4n+ +(λ 1 λ ) n+1 λ 1 λ = B 4n+3 +1
6 49 P.K. Ray Theorem.1. For n 1, we have B 4n+3 +1 = 4B n+1 C n+1 C n+1. Proof. SubstitutingB n+ = B n+1 C n+1 fromlemma.5intolemma.11, we obtain the desired result. Theorem.13. For n 1, the following identity is valid: Proof. By (.1), we have C 4n+3 +3 = C n+1 C n+. C n+1 C n+ = λn+1 1 +λ n+1 λ n+ which completes the proof. = λ4n+3 1 +λ 4n+3 = C 4n λ n+ +(λ 1 λ ) n+1λ 1 +λ Theorem.14. For n 1, the following identity is valid: C 4n+3 3 = 16B n+1 B n+. Proof. Since (λ 1 λ ) = 3, we obtain which ends the proof. 16B n+1 B n+ = λn+1 1 λ n+1 λ 1 λ ) = λ4n+3 1 +λ 4n+3 = C 4n+3 3 λ n+ 1 λ n+ λ 1 λ (λ 1 λ ) n+1λ 1 +λ The following corollary is an immediate consequence of Theorem.14. Corollary.15. For n 1, we have C 4n+3 3 = 3B n+1 C n+1 B n+1.
7 NEW IDENTITIES FOR THE COMMON FACTORS OF References [1] A. Behera and G.K. Panda, On the square roots of triangular numbers, The Fibonacci Quarterly, 37, No. (1999), [] A. Berczes, K. Liptai, I. Pink, On generalized balancing numbers, Fibonacci Quarterly, 48, No. (010), [3] R. Keskin, O. Karaatly, Some new properties of balancing numbers and square triangular numbers, Journal of Integer Sequences, 15, No. 1 (01). [4] K. Liptai, Fibonacci balancing numbers, The Fibonacci Quarterly, 4, No. 4 (004), [5] K. Liptai, Lucas balancing numbers, Acta Math.Univ. Ostrav, 14, No. 1 (006), [6] K. Liptai, F. Luca, A. Pinter, L. Szalay, Generalized balancing numbers, Indagationes Math. N. S., 0 (009), [7] P. Olajos, Properties of balancing, cobalancing and generalized balancing numbers, Annales Mathematicae et Informaticae, 37 (010), [8] G.K. Panda, P.K. Ray, Some links of balancing and cobalancing numbers with Pell and associated Pell numbers, Bulletin of the Institute of Mathematics, Academia Sinica (New Series), 6, No. 1 (011), [9] G.K. Panda, P.K. Ray, Cobalancing numbers and cobalancers, International Journal of Mathematics and Mathematical Sciences, 8 (005), [10] G.K. Panda, Some fascinating properties of balancing numbers, Proc. Eleventh Internat. Conference on Fibonacci Numbers and Their Applications, Cong. Numerantium, 194 (009), [11] P.K. Ray, Application of Chybeshev polynomials in factorization of balancing and Lucasbalancing numbers, Bol. Soc. Paran. Mat., 30, No. (01), [1] P.K. Ray, Factorization of negatively subscripted balancing and Lucasbalancing numbers, Bol.Soc.Paran.Mat., 31, No. (013), [13] P.K. Ray, Curious congruences for balancing numbers, Int. J. Contemp. Sciences, 7, No. 18 (01),
8 494 P.K. Ray [14] P.K. Ray, Certain matrices associated with balancing and Lucasbalancing numbers, Matematika, 8, No. 1 (01), 15.
On Linear Recursive Sequences with Coefficients in ArithmeticGeometric Progressions
Applied Mathematical Sciences, Vol. 9, 015, no. 5, 595607 HIKARI Ltd, www.mhikari.com http://dx.doi.org/10.1988/ams.015.5163 On Linear Recursive Sequences with Coefficients in ArithmeticGeometric Progressions
More informationGENERALIZED FIBONACCI AND LUCAS NUMBERS OF THE FORM wx 2 AND wx 2 1
Bull. Korean Math. Soc. 51 (2014), No. 4, pp. 1041 1054 http://dx.doi.org/10.4134/bkms.2014.51.4.1041 GENERALIZED FIBONACCI AND LUCAS NUMBERS OF THE FORM wx 2 AND wx 2 1 Ref ik Kesk in Abstract. Let P
More information#A2 INTEGERS 12A (2012): John Selfridge Memorial Issue ON A CONJECTURE REGARDING BALANCING WITH POWERS OF FIBONACCI NUMBERS
#A2 INTEGERS 2A (202): John Selfridge Memorial Issue ON A CONJECTURE REGARDING BALANCING WITH POWERS OF FIBONACCI NUMBERS Saúl Díaz Alvarado Facultad de Ciencias, Universidad Autónoma del Estado de México,
More informationOn Generalized kfibonacci Sequence by TwoCrossTwo Matrix
Global Journal of Mathematical Analysis, 5 () (07) 5 Global Journal of Mathematical Analysis Website: www.sciencepubco.com/index.php/gjma doi: 0.449/gjma.v5i.6949 Research paper On Generalized kfibonacci
More informationABSTRACT. In this note, we find all the solutions of the Diophantine equation x k = y n, 1, y 1, k N, n INTRODUCTION
Florian Luca Instituto de Matemáticas UNAM, Campus Morelia Apartado Postal 273 (Xangari), C.P. 58089, Morelia, Michoacán, Mexico email: fluca@matmor.unam.mx Alain Togbé Mathematics Department, Purdue
More informationOn second order nonhomogeneous recurrence relation
Annales Mathematicae et Informaticae 41 (2013) pp. 20 210 Proceedings of the 1 th International Conference on Fibonacci Numbers and Their Applications Institute of Mathematics and Informatics, Eszterházy
More informationPAijpam.eu THE PERIOD MODULO PRODUCT OF CONSECUTIVE FIBONACCI NUMBERS
International Journal of Pure and Applied Mathematics Volume 90 No. 014, 544 ISSN: 1118080 (printed version); ISSN: 11495 (online version) url: http://www.ipam.eu doi: http://dx.doi.org/10.17/ipam.v90i.7
More informationFibonacci Diophantine Triples
Fibonacci Diophantine Triples Florian Luca Instituto de Matemáticas Universidad Nacional Autonoma de México C.P. 58180, Morelia, Michoacán, México fluca@matmor.unam.mx László Szalay Institute of Mathematics
More informationFibonacci and Lucas numbers via the determinants of tridiagonal matrix
Notes on Number Theory and Discrete Mathematics Print ISSN 30 532, Online ISSN 2367 8275 Vol 24, 208, No, 03 08 DOI: 07546/nntdm208240308 Fibonacci and Lucas numbers via the determinants of tridiagonal
More informationGeneralized Identities on Products of FibonacciLike and Lucas Numbers
Generalized Identities on Products of FibonacciLike and Lucas Numbers Shikha Bhatnagar School of Studies in Mathematics, Vikram University, Ujjain (M P), India suhani_bhatnagar@rediffmailcom Omrakash
More informationON SUMS OF SQUARES OF PELLLUCAS NUMBERS. Gian Mario Gianella University of Torino, Torino, Italy, Europe.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A15 ON SUMS OF SQUARES OF PELLLUCAS NUMBERS Zvonko Čerin University of Zagreb, Zagreb, Croatia, Europe cerin@math.hr Gian Mario Gianella
More informationTHE LOGBEHAVIOR OF THE SEQUENCE FOR THE PARTIAL SUM OF A LOGCONVEX SEQUENCE. 1. Introduction
SARAJEVO JOURNAL OF MATHEMATICS Vol.13 (26), No.2, (2017), 163 178 DOI: 10.5644/SJM.13.2.04 THE LOGBEHAVIOR OF THE SEQUENCE FOR THE PARTIAL SUM OF A LOGCONVEX SEQUENCE FENGZHEN ZHAO Abstract. In this
More informationPAijpam.eu NUMERICAL SOLUTION OF WAVE EQUATION USING HAAR WAVELET Inderdeep Singh 1, Sangeeta Arora 2, Sheo Kumar 3
International Journal of Pure and Applied Mathematics Volume 98 No. 4 25, 457469 ISSN: 388 (printed version); ISSN: 343395 (online version) url: http://www.ijpam.eu doi: http://dx.doi.org/.2732/ijpam.v98i4.4
More informationON POLYNOMIAL IDENTITIES FOR RECURSIVE SEQUENCES
Miskolc Mathematical Notes HU eissn 17872413 Vol. 18 (2017), No. 1, pp. 327 336 DOI: 10.18514/MMN.2017.1776 ON POLYNOMIAL IDENTITIES FOR RECURSIVE SEQUENCES I. MARTINJAK AND I. VRSALJKO Received 09 September,
More informationSTRONGLY SEMICOMMUTATIVE RINGS RELATIVE TO A MONOID. Ayoub Elshokry 1, Eltiyeb Ali 2. Northwest Normal University Lanzhou , P.R.
International Journal of Pure and Applied Mathematics Volume 95 No. 4 2014, 611622 ISSN: 13118080 printed version); ISSN: 13143395 online version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v95i4.14
More informationDeterminant and Permanent of Hessenberg Matrix and Fibonacci Type Numbers
Gen. Math. Notes, Vol. 9, No. 2, April 2012, pp.3241 ISSN 22197184; Copyright c ICSRS Publication, 2012 www.icsrs.org Available free online at http://www.geman.in Determinant and Permanent of Hessenberg
More informationCOMMON FIXED POINT THEOREMS OF WEAK RECIPROCAL CONTINUITY IN METRIC SPACES. Saurabh Manro 1, Sanjay Kumar 2, Satwinder Singh Bhatia 3, Shin Min Kang 4
International Journal of Pure and Applied Mathematics Volume 88 No. 2 2013, 297304 ISSN: 13118080 (printed version); ISSN: 13143395 (online version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v88i2.11
More informationarxiv: v1 [math.nt] 29 Dec 2017
arxiv:1712.10138v1 [math.nt] 29 Dec 2017 On the Diophantine equation F n F m = 2 a Zafer Şiar a and Refik Keskin b a Bingöl University, Mathematics Department, Bingöl/TURKEY b Sakarya University, Mathematics
More informationPower of Two Classes in k Generalized Fibonacci Sequences
Revista Colombiana de Matemáticas Volumen 482014)2, páginas 219234 Power of Two Classes in k Generalized Fibonacci Sequences Clases de potencias de dos en sucesiones k generalizadas de Fibonacci Carlos
More informationSome Generalized Fibonomial Sums related with the Gaussian q Binomial sums
Bull. Math. Soc. Sci. Math. Roumanie Tome 55(103 No. 1, 01, 51 61 Some Generalized Fibonomial Sums related with the Gaussian q Binomial sums by Emrah Kilic, Iler Aus and Hideyui Ohtsua Abstract In this
More informationTILING PROOFS OF SOME FORMULAS FOR THE PELL NUMBERS OF ODD INDEX
#A05 INTEGERS 9 (2009), 5364 TILING PROOFS OF SOME FORMULAS FOR THE PELL NUMBERS OF ODD INDEX Mark Shattuck Department of Mathematics, University of Tennessee, Knoxville, TN 379961300 shattuck@math.utk.edu
More information1 Introduction. 2 Determining what the J i blocks look like. December 6, 2006
Jordan Canonical Forms December 6, 2006 1 Introduction We know that not every n n matrix A can be diagonalized However, it turns out that we can always put matrices A into something called Jordan Canonical
More informationN. Karthikeyan 1, N. Rajesh 2. Jeppiaar Engineering College Chennai, , Tamilnadu, INDIA 2 Department of Mathematics
International Journal of Pure and Applied Mathematics Volume 103 No. 1 2015, 1926 ISSN: 13118080 (printed version); ISSN: 13143395 (online version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v103i1.2
More informationIDEALS AND THEIR FUZZIFICATIONS IN IMPLICATIVE SEMIGROUPS
International Journal of Pure and Applied Mathematics Volume 104 No. 4 2015, 543549 ISSN: 13118080 (printed version); ISSN: 13143395 (online version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v104i4.6
More informationPAijpam.eu CONVOLUTIONAL CODES DERIVED FROM MELAS CODES
International Journal of Pure and Applied Mathematics Volume 85 No. 6 013, 10011008 ISSN: 13118080 (printed version); ISSN: 13143395 (online version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.173/ijpam.v85i6.3
More informationPAijpam.eu COMMON FIXED POINT THEOREMS OF COMPATIBLE MAPPINGS IN METRIC SPACES
International Journal of Pure and Applied Mathematics Volume 84 No. 203, 783 ISSN: 38080 (printed version); ISSN: 343395 (online version) url: http://www.ijpam.eu doi: http://dx.doi.org/0.2732/ijpam.v84i.3
More informationFAREYPELL SEQUENCE, APPROXIMATION TO IRRATIONALS AND HURWITZ S INEQUALITY (COMMUNICATED BY TOUFIK MANSOUR)
Bulletin of Mathematical Analysis Applications ISSN: 89 URL: http://wwwmathaag Volume 8 Issue (06) Pages  FAREYPELL SEQUENCE APPROXIMATION TO IRRATIONALS AND HURWITZ S INEQUALITY (COMMUNICATED BY TOUFIK
More informationExact Determinants of the RFPrLrR Circulant Involving Jacobsthal, JacobsthalLucas, Perrin and Padovan Numbers
Tingting Xu Zhaolin Jiang Exact Determinants of the RFPrLrR Circulant Involving Jacobsthal JacobsthalLucas Perrin and Padovan Numbers TINGTING XU Department of Mathematics Linyi University Shuangling
More informationOn integer solutions to x 2 dy 2 = 1, z 2 2dy 2 = 1
ACTA ARITHMETICA LXXXII.1 (1997) On integer solutions to x 2 dy 2 = 1, z 2 2dy 2 = 1 by P. G. Walsh (Ottawa, Ont.) 1. Introduction. Let d denote a positive integer. In [7] Ono proves that if the number
More informationCounting Palindromic Binary Strings Without rruns of Ones
1 3 47 6 3 11 Journal of Integer Sequences, Vol. 16 (013), Article 13.8.7 Counting Palindromic Binary Strings Without rruns of Ones M. A. Nyblom School of Mathematics and Geospatial Science RMIT University
More informationsgeneralized Fibonacci Numbers: Some Identities, a Generating Function and Pythagorean Triples
International Journal of Mathematical Analysis Vol. 8, 2014, no. 36, 17571766 HIKARI Ltd, www.mhikari.com http://dx.doi.org/10.12988/ijma.2014.47203 sgeneralized Fibonacci Numbers: Some Identities,
More informationCarmen s Core Concepts (Math 135)
Carmen s Core Concepts (Math 135) Carmen Bruni University of Waterloo Week 4 1 Principle of Mathematical Induction 2 Example 3 Base Case 4 Inductive Hypothesis 5 Inductive Step When Induction Isn t Enough
More informationExplicit solution of a class of quartic Thue equations
ACTA ARITHMETICA LXIV.3 (1993) Explicit solution of a class of quartic Thue equations by Nikos Tzanakis (Iraklion) 1. Introduction. In this paper we deal with the efficient solution of a certain interesting
More informationCullen Numbers in Binary Recurrent Sequences
Cullen Numbers in Binary Recurrent Sequences Florian Luca 1 and Pantelimon Stănică 2 1 IMATEUNAM, Ap. Postal 613 (Xangari), CP 58 089 Morelia, Michoacán, Mexico; email: fluca@matmor.unam.mx 2 Auburn
More informationOn cycle index and orbit stabilizer of Symmetric group
The International Journal Of Engineering And Science (IJES) Volume 3 Issue 1 Pages 1826 2014 ISSN (e): 2319 1813 ISSN (p): 2319 1805 On cycle index and orbit stabilizer of Symmetric group 1 Mogbonju M.
More informationIntroduction to Number Theory
INTRODUCTION Definition: Natural Numbers, Integers Natural numbers: N={0,1,, }. Integers: Z={0,±1,±, }. Definition: Divisor If a Z can be writeen as a=bc where b, c Z, then we say a is divisible by b or,
More informationWhen do Fibonacci invertible classes modulo M form a subgroup?
Calhoun: The NPS Institutional Archive DSace Reository Faculty and Researchers Faculty and Researchers Collection 2013 When do Fibonacci invertible classes modulo M form a subgrou? Luca, Florian Annales
More informationOn h(x)fibonacci octonion polynomials
Alabama Journal of Mathematics 39 (05) ISSN 3730404 On h(x)fibonacci octonion polynomials Ahmet İpek Karamanoğlu Mehmetbey University, Science Faculty of Kamil Özdağ, Department of Mathematics, Karaman,
More informationPell Equation x 2 Dy 2 = 2, II
Irish Math Soc Bulletin 54 2004 73 89 73 Pell Equation x 2 Dy 2 2 II AHMET TEKCAN Abstract In this paper solutions of the Pell equation x 2 Dy 2 2 are formulated for a positive nonsquare integer D using
More informationOn a certain vector crank modulo 7
On a certain vector crank modulo 7 Michael D Hirschhorn School of Mathematics and Statistics University of New South Wales Sydney, NSW, 2052, Australia mhirschhorn@unsweduau Pee Choon Toh Mathematics &
More informationNONABELIAN GROUPS WITH PERFECT ORDER SUBSETS
NONABELIAN GROUPS WITH PERFECT ORDER SUBSETS CARRIE E. FINCH AND LENNY JONES Abstract. Let G be a finite group and let x G. Define the order subset of G determined by x to be the set of all elements in
More informationINVERSE PROBLEMS FOR STURMLIOUVILLE OPERATORS WITH BOUNDARY CONDITIONS DEPENDING ON A SPECTRAL PARAMETER
Electronic Journal of Differential Equations, Vol. 217 (217), No. 26, pp. 1 7. ISSN: 1726691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu INVERSE PROBLEMS FOR STURMLIOUVILLE OPERATORS
More informationDiophantine Equations Concerning Linear Recurrences
Diophantine Equations Concerning Linear Recurrences PhD thesis László Szalay Lajos Kossuth University Debrecen, 1999. Contents 1 Introduction 1 2 Notation and basic results 6 2.1 Recurrence sequences.......................
More informationIs there a computable upper bound for the height of a solution of a Diophantine equation with a unique solution in positive integers?
Is there a computable upper bound for the height of a solution of a Diophantine equation with a unique solution in positive integers? Apoloniusz Tyszka arxiv:1404.5975v18 [math.lo] 6 Apr 2017 Abstract
More informationFIXED POINT THEOREM IN STRUCTURE FUZZY METRIC SPACE
International Journal of Pure and Applied Mathematics Volume 99 No. 3 2015, 367371 ISSN: 13118080 (printed version); ISSN: 13143395 (online version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v99i3.11
More informationMinimal multipliers for consecutive Fibonacci numbers
ACTA ARITHMETICA LXXV3 (1996) Minimal multipliers for consecutive Fibonacci numbers by K R Matthews (Brisbane) 1 Introduction The Fibonacci and Lucas numbers F n, L n (see [2]) are defined by Since F 1
More informationMaximum unionfree subfamilies
Maximum unionfree subfamilies Jacob Fox Choongbum Lee Benny Sudakov Abstract An old problem of Moser asks: how large of a unionfree subfamily does every family of m sets have? A family of sets is called
More informationarxiv: v1 [math.co] 10 Nov 2016
Indecomposable 1factorizations of the complete multigraph λk 2n for every λ 2n arxiv:1611.03221v1 [math.co] 10 Nov 2016 S. Bonvicini, G. Rinaldi November 11, 2016 Abstract A 1factorization of the complete
More informationarxiv: v3 [math.co] 6 Aug 2016
ANALOGUES OF A FIBONACCILUCAS IDENTITY GAURAV BHATNAGAR arxiv:1510.03159v3 [math.co] 6 Aug 2016 Abstract. Sury s 2014 proof of an identity for Fibonacci and Lucas numbers (Identity 236 of Benjamin and
More informationGeneralized Bivariate Lucas ppolynomials and Hessenberg Matrices
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 15 (2012), Article 12.3.4 Generalized Bivariate Lucas ppolynomials and Hessenberg Matrices Kenan Kaygisiz and Adem Şahin Department of Mathematics Faculty
More informationThe kfibonacci matrix and the Pascal matrix
Cent Eur J Math 9(6 0 40340 DOI: 0478/s533000899 Central European Journal of Mathematics The Fibonacci matrix and the Pascal matrix Research Article Sergio Falcon Department of Mathematics and Institute
More informationTILING ABELIAN GROUPS WITH A SINGLE TILE
TILING ABELIAN GROUPS WITH A SINGLE TILE S.J. EIGEN AND V.S. PRASAD Abstract. Suppose G is an infinite Abelian group that factorizes as the direct sum G = A B: i.e., the Btranslates of the single tile
More informationA Remark on Prime Divisors of Lengths of Sides of Heron Triangles
A Remark on Prime Divisors of Lengths of Sides of Heron Triangles István Gaál, István Járási, Florian Luca CONTENTS 1. Introduction 2. Proof of Theorem 1.1 3. Proof of Theorem 1.2 4. Proof of Theorem 1.3
More informationSOME PÓLYATYPE IRREDUCIBILITY CRITERIA FOR MULTIVARIATE POLYNOMIALS NICOLAE CIPRIAN BONCIOCAT, YANN BUGEAUD, MIHAI CIPU, AND MAURICE MIGNOTTE
SOME PÓLYATYPE IRREDUCIBILITY CRITERIA FOR MULTIVARIATE POLYNOMIALS NICOLAE CIPRIAN BONCIOCAT, YANN BUGEAUD, MIHAI CIPU, AND MAURICE MIGNOTTE Abstract. We provide irreducibility criteria for multivariate
More informationSTRONG NORMALITY AND GENERALIZED COPELAND ERDŐS NUMBERS
#A INTEGERS 6 (206) STRONG NORMALITY AND GENERALIZED COPELAND ERDŐS NUMBERS Elliot Catt School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, New South Wales, Australia
More informationCool theorems proved by undergraduates
Cool theorems proved by undergraduates Ken Ono Emory University Child s play... Child s play... Thanks to the, Child s play... Thanks to the, undergrads and I play by... Child s play... Thanks to the,
More informationON THE REDUCIBILITY OF EXACT COVERING SYSTEMS
ON THE REDUCIBILITY OF EXACT COVERING SYSTEMS OFIR SCHNABEL arxiv:1402.3957v2 [math.co] 2 Jun 2015 Abstract. There exist irreducible exact covering systems (ECS). These are ECS which are not a proper split
More informationAlmost perfect powers in consecutive integers (II)
Indag. Mathem., N.S., 19 (4), 649 658 December, 2008 Almost perfect powers in consecutive integers (II) by N. Saradha and T.N. Shorey School of Mathematics, Tata Institute of Fundamental Research, Homi
More informationALTERNATING SUMS OF FIBONACCI PRODUCTS
ALTERNATING SUMS OF FIBONACCI PRODUCTS ZVONKO ČERIN Abstract. We consider alternating sums of squares of odd even terms of the Fibonacci sequence alternating sums of their products. These alternating sums
More informationCONVOLUTION TREES AND PASCALT TRIANGLES. JOHN C. TURNER University of Waikato, Hamilton, New Zealand (Submitted December 1986) 1.
JOHN C. TURNER University of Waikato, Hamilton, New Zealand (Submitted December 986). INTRODUCTION Pascal (666) made extensive use of the famous arithmetical triangle which now bears his name. He wrote
More informationModular Arithmetic Instructor: Marizza Bailey Name:
Modular Arithmetic Instructor: Marizza Bailey Name: 1. Introduction to Modular Arithmetic If someone asks you what day it is 145 days from now, what would you answer? Would you count 145 days, or find
More informationGROWTH OF THE MAXIMUM MODULUS OF POLYNOMIALS WITH PRESCRIBED ZEROS M. S. PUKHTA. 1. Introduction and Statement of Result
Journal of Classical Analysis Volume 5, Number 2 (204, 07 3 doi:0.753/jca0509 GROWTH OF THE MAXIMUM MODULUS OF POLYNOMIALS WITH PRESCRIBED ZEROS M. S. PUKHTA Abstract. If p(z n a j j is a polynomial
More information1. INTRODUCTION. Ll5F 2 = 4(l)" (1.1)
Ray Melham School of Mathematical Sciences, University of Technology, Sydney PO Box 123, Broadway, NSW 2007, Australia (Submitted April 1997) Long [4] considered the identity 1. INTRODUCTION Ll5F 2 =
More informationarxiv: v1 [math.nt] 8 Sep 2014
arxiv:1409.2463v1 [math.nt] 8 Sep 2014 On the Diophantine equation X 2N +2 2α 5 2β p 2γ = Z 5 Eva G. Goedhart and Helen G. Grundman Abstract We prove that for each odd prime p, positive integer α, and
More informationarxiv: v1 [math.nt] 9 Sep 2017
arxiv:179.2954v1 [math.nt] 9 Sep 217 ON THE FACTORIZATION OF x 2 +D GENERALIZED RAMANUJANNAGELL EQUATION WITH HUGE SOLUTION) AMIR GHADERMARZI Abstract. Let D be a positive nonsquare integer such that
More informationBridges for concatenation hierarchies
Bridges for concatenation hierarchies JeanÉric Pin LIAFA, CNRS and Université Paris VII 2 Place Jussieu 75251 Paris Cedex O5, FRANCE email: JeanEric.Pin@liafa.jussieu.fr Abstract. In the seventies,
More informationON SUMS OF SQUARES OF ODD AND EVEN TERMS OF THE LUCAS SEQUENCE
ON SUMS OF SQUARES OF ODD AND EVEN TERMS OF THE LUCAS SEQUENCE ZVONKO ČERIN 1. Introduction The Fibonacci and Lucas sequences F n and L n are defined by the recurrence relations and F 1 = 1, F 2 = 1, F
More informationPRIMITIVE PRIME FACTORS IN SECONDORDER LINEAR RECURRENCE SEQUENCES
PRIMITIVE PRIME FACTORS IN SECONDORDER LINEAR RECURRENCE SEQUENCES Andrew Granville To Andrzej Schinzel on his 75th birthday, with thanks for the many inspiring papers Abstract. For a class of Lucas sequences
More informationJoshua N. Cooper Department of Mathematics, Courant Institute of Mathematics, New York University, New York, NY 10012, USA.
CONTINUED FRACTIONS WITH PARTIAL QUOTIENTS BOUNDED IN AVERAGE Joshua N. Cooper Department of Mathematics, Courant Institute of Mathematics, New York University, New York, NY 10012, USA cooper@cims.nyu.edu
More informationRecursion. Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry. Fall 2007
Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Fall 2007 1 / 47 Computer Science & Engineering 235 to Discrete Mathematics Sections 7.17.2 of Rosen Recursive Algorithms 2 / 47 A recursive
More informationTHE padic VALUATION OF LUCAS SEQUENCES
THE padic VALUATION OF LUCAS SEQUENCES CARLO SANNA Abstract. Let (u n) n 0 be a nondegenerate Lucas sequence with characteristic polynomial X 2 ax b, for some relatively prime integers a and b. For each
More informationMAPPING AND ITS APPLICATIONS. J. Jeyachristy Priskillal 1, P. Thangavelu 2
International Journal of Pure and Applied Mathematics Volume 11 No. 1 017, 1771 ISSN: 131100 (printed version); ISSN: 13143395 (online version) url: http://www.ijpam.eu doi: 10.173/ijpam.v11i1.14 PAijpam.eu
More informationCHOLESKY ALGORITHM MATRICES OF FIBONACCI TYPE AND PROPERTIES OF GENERALIZED SEQUENCES*
CHOLESKY ALGORITHM MATRICES OF FIBONACCI TYPE AND PROPERTIES OF GENERALIZED SEQUENCES* Alwyn F. Horadam University of New England, Armidale, Australia Piero FilipponI Fondazione Ugo Bordoni, Rome, Italy
More informationTHE EXISTENCE OF COMMON FIXED POINTS FOR FAINTLY COMPATIBLE MAPPINGS IN MENGER SPACES
International Journal of Pure and Applied Mathematics Volume 97 No. 1 2014, 3748 ISSN: 13118080 (printed version); ISSN: 13143395 (online version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v97i1.5
More informationPAijpam.eu OBTAINING A COMPROMISE SOLUTION OF A MULTI OBJECTIVE FIXED CHARGE PROBLEM IN A FUZZY ENVIRONMENT
International Journal of Pure and Applied Mathematics Volume 98 No. 2 2015, 193210 ISSN: 13118080 (printed version); ISSN: 13143395 (online version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v98i2.3
More informationTwo Diophantine Approaches to the Irreducibility of Certain Trinomials
Two Diophantine Approaches to the Irreducibility of Certain Trinomials M. Filaseta 1, F. Luca 2, P. Stănică 3, R.G. Underwood 3 1 Department of Mathematics, University of South Carolina Columbia, SC 29208;
More informationGOLDEN SEQUENCES OF MATRICES WITH APPLICATIONS TO FIBONACCI ALGEBRA
GOLDEN SEQUENCES OF MATRICES WITH APPLICATIONS TO FIBONACCI ALGEBRA JOSEPH ERCOLANO Baruch College, CUNY, New York, New York 10010 1. INTRODUCTION As is well known, the problem of finding a sequence of
More informationAn Application of Matricial Fibonacci Identities to the Computation of Spectral Norms
An Application of Matricial Fibonacci Identities to the Computation of Spectral Norms John Dixon, Ben Mathes, and David Wheeler February, 01 1 Introduction Among the most intensively studied integer sequences
More informationExploration of Fibonacci function Prof. K. Raja Rama Gandhi
Bulletin of Mathematical Sciences and Applications Online: 20120801 ISSN: 22789634, Vol. 1, pp 5762 doi:10.18052/www.scipress.com/bmsa.1.57 2012 SciPress Ltd., Switzerland Keywords: Fibonacci function,
More informationOn the Exponents of NonTrivial Divisors of Odd Numbers and a Generalization of Proth s Primality Theorem
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.7 On the Exponents of NonTrivial Divisors of Odd Numbers and a Generalization of Proth s Primality Theorem Tom Müller Institut
More informationAndrew Granville To Andrzej Schinzel on his 75th birthday, with thanks for the many inspiring papers
PRIMITIVE PRIME FACTORS IN SECONDORDER LINEAR RECURRENCE SEQUENCES Andrew Granville To Andrzej Schinzel on his 75th birthday, with thanks for the many inspiring papers Abstract. For a class of Lucas sequences
More informationTable of Contents. 2013, Pearson Education, Inc.
Table of Contents Chapter 1 What is Number Theory? 1 Chapter Pythagorean Triples 5 Chapter 3 Pythagorean Triples and the Unit Circle 11 Chapter 4 Sums of Higher Powers and Fermat s Last Theorem 16 Chapter
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics ON THE DIOPHANTINE EQUATION xn 1 x 1 = y Yann Bugeaud, Maurice Mignotte, and Yves Roy Volume 193 No. 2 April 2000 PACIFIC JOURNAL OF MATHEMATICS Vol. 193, No. 2, 2000 ON
More informationResistance distance in wheels and fans
Resistance distance in wheels and fans R B Bapat Somit Gupta February 4, 009 Abstract: The wheel graph is the join of a single vertex and a cycle, while the fan graph is the join of a single vertex and
More informationConstruction of `Wachspress type' rational basis functions over rectangles
Proc. Indian Acad. Sci. (Math. Sci.), Vol. 110, No. 1, February 2000, pp. 69±77. # Printed in India Construction of `Wachspress type' rational basis functions over rectangles P L POWAR and S S RANA Department
More informationON THE EXTENSIBILITY OF DIOPHANTINE TRIPLES {k 1, k + 1, 4k} FOR GAUSSIAN INTEGERS. Zrinka Franušić University of Zagreb, Croatia
GLASNIK MATEMATIČKI Vol. 43(63)(2008), 265 291 ON THE EXTENSIBILITY OF DIOPHANTINE TRIPLES {k 1, k + 1, 4k} FOR GAUSSIAN INTEGERS Zrinka Franušić University of Zagreb, Croatia Abstract. In this paper,
More informationA NOTE ON THE DIOPHANTINE EQUATION a x b y =c
MATH. SCAND. 107 010, 161 173 A NOTE ON THE DIOPHANTINE EQUATION a x b y =c BO HE, ALAIN TOGBÉ and SHICHUN YANG Abstract Let a,b, and c be positive integers. We show that if a, b = N k 1,N, where N,k,
More informationJ. Sanabria, E. Acosta, M. SalasBrown and O. García
F A S C I C U L I M A T H E M A T I C I Nr 54 2015 DOI:10.1515/fascmath20150009 J. Sanabria, E. Acosta, M. SalasBrown and O. García CONTINUITY VIA Λ I OPEN SETS Abstract. Noiri and Keskin [8] introduced
More informationLacunary Riesz χ 3 R λmi. Key Words: Analytic sequence, Orlicz function, Double sequences, Riesz space, Riesz convergence, Pringsheim convergence.
Bol Soc Paran Mat (3s) v 37 2 (209): 29 44 c SPM ISSN27588 on line ISSN0037872 in press SPM: wwwspmuembr/bspm doi:05269/bspmv37i23430 ) Triple Almost Lacunary Riesz Sequence Spaces µ nl Defined by Orlicz
More informationA Generalization of Bernoulli's Inequality
Florida International University FIU Digital Commons Department of Mathematics and Statistics College of Arts, Sciences & Education 200 A Generalization of Bernoulli's Inequality Laura De Carli Department
More informationParity Dominating Sets in Grid Graphs
Parity Dominating Sets in Grid Graphs John L. Goldwasser and William F. Klostermeyer Dept. of Mathematics West Virginia University Morgantown, WV 26506 Dept. of Computer and Information Sciences University
More informationRANKINCOHEN BRACKETS AND VAN DER POLTYPE IDENTITIES FOR THE RAMANUJAN S TAU FUNCTION
RANKINCOHEN BRACKETS AND VAN DER POLTYPE IDENTITIES FOR THE RAMANUJAN S TAU FUNCTION B. RAMAKRISHNAN AND BRUNDABAN SAHU Abstract. We use RankinCohen brackets for modular forms and quasimodular forms
More informationPOLYGONALSIERPIŃSKIRIESEL SEQUENCES WITH TERMS HAVING AT LEAST TWO DISTINCT PRIME DIVISORS
#A40 INTEGERS 16 (2016) POLYGONALSIERPIŃSKIRIESEL SEQUENCES WITH TERMS HAVING AT LEAST TWO DISTINCT PRIME DIVISORS Daniel Baczkowski Department of Mathematics, The University of Findlay, Findlay, Ohio
More informationEVERY NATURAL NUMBER IS THE SUM OF FORTYNINE PALINDROMES
#A3 INTEGERS 16 (2016) EVERY NATURAL NUMBER IS THE SUM OF FORTYNINE PALINDROMES William D. Banks Department of Mathematics, University of Missouri, Columbia, Missouri bankswd@missouri.edu Received: 9/4/15,
More informationA PROOF OF A CONJECTURE OF MELHAM
A PROOF OF A CONJECTRE OF MELHAM EMRAH KILIC, ILKER AKKS, AND HELMT PRODINGER Abstract. In this paper, we consider Melha s conecture involving Fibonacci and Lucas nubers. After rewriting it in ters of
More informationNew congruences for overcubic partition pairs
New congruences for overcubic partition pairs M. S. Mahadeva Naika C. Shivashankar Department of Mathematics, Bangalore University, Central College Campus, Bangalore560 00, Karnataka, India Department
More informationA Note on Inextensible Flows of Partially & Pseudo Null Curves in E 4 1
Prespacetime Journal April 216 Volume 7 Issue 5 pp. 818 827 818 Article A Note on Inextensible Flows of Partially & Pseudo Null Curves in E 4 1 Zühal Küçükarslan Yüzbaşı 1 & & Mehmet Bektaş Firat University,
More informationSOME GENERALIZATION OF MINTY S LEMMA. DooYoung Jung
J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math. 6(1999), no 1. 33 37 SOME GENERALIZATION OF MINTY S LEMMA DooYoung Jung Abstract. We obtain a generalization of Behera and Panda s result on nonlinear
More informationOrdering trees by their largest eigenvalues
Linear Algebra and its Applications 370 (003) 75 84 www.elsevier.com/locate/laa Ordering trees by their largest eigenvalues An Chang a,, Qiongxiang Huang b a Department of Mathematics, Fuzhou University,
More information1. INTRODUCTION. n=. A few applications are also presented at the end of the Note. The following Theorem is established in the present Note:
CONVOLVING THE mth POWERS OF THE CONSECUTIVE INTEGERS WITH THE GENERAL FIBONACCI SEQUENCE USING CARLITZ S WEIGHTED STIRLING POLYNOMIALS OF THE SECOND KIND N. Gauthier Department of Physics, The Royal
More information