Some congruences concerning second order linear recurrences

Size: px
Start display at page:

Download "Some congruences concerning second order linear recurrences"

Transcription

1 Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae,. (1997) pp Some congruences concerning second order linear recurrences JAMES P. JONES PÉTER KISS Abstract. Let U n V n (n=0,1,,...) be sequences of integers satisfying a second order linear recurrence relation with initial terms U 0 =0, U 1 =1, V 0 =, V 1 =A. In this paper we investigate the congruence properties of the terms U nk V nk, where the moduli are powers of U n V n. Let U n V n (n = 0,1,,...) be second order linear recursive sequences of integers defined by U n = AU n 1 BU n (n > 1) V n = AV n 1 BV n (n > 1), where A B are nonzero rational integers the initial terms are U 0 = 0, U 1 = 1, V 0 =, V 1 = A. enote by α,β the roots of the characteristic equation x Ax + B = 0 suppose = A 4B 0 hence that α β. In this case, as it is well known, the terms of the sequences can be expressed as (1) U n = αn β n α β V n = α n + β n for any n 0. Many identities congruence properties are known for the sequences U n V n (see, e.g. [1], [4], [5] [6]). Some congruence properties are also known when the modulus is a power of a term of the sequences (see [], [3], [7] [8]). In [3] we derived some congruences where the moduli was Un 3, V n or V n 3. Among other congruences we proved that U nk kb n k 1 Un (mod U 3 n) Research supported by the Hungarian National Research Science Foundation, Operating Grant Number OTKA T

2 30 James P. Jones Péter Kiss when k is odd a similar congruence for even k. In this paper we extend the results of [3]. We derive congruences in which the moduli are product of higher powers of U n V n. Theorem. Let U n V n be second order linear recurrences defined above let = A 4B be the discriminant of the characteristic equation. Then for positive integers n k we have 1. U nk kb k 1 n U n + k(k 1) B k 3 n U 3 n (mod U 5 n ), k odd,. U nk k B k n V n U n + k(k 4) 48 B k 4 n V n Un 3 (mod V n Un 5 ), k even, 3. V nk k( 1) k 1 B k 1 n V n + k(k 1) ( 1) k 3 B k 3 n Vn 3 (mod Vn 5 ), k odd, 4. V nk ( 1) k B k n + k 4 k ( 1) B k n V n (mod V 4 n ), k even, 5. U nk U n ( 1) k 1 B k 1 n + k 1 8 ( 1) k 3 B k 3 n U n V n (mod U n V 4 n ), k odd, 6. U nk k k ( 1) B k n U n V n + k(k 4) 48 ( 1) k 4 B k 4 n U n Vn 3 (mod U nvn 5 ), k even, 7. V nk B k 1 n V n + k 1 8 B k 3 n V n Un (mod V n Un 4 ), k odd, 8. V nk B k n + k 4 B k n U n (mod U 4 n ), k even. We note that the congruences of [3] follow as consequences of this theorem. For the proof of the Theorem we need some auxiliary results which are known (see e.g. [6]) but we show short proofs for them. In the followings we suppose that A > 0 hence that α = A + β = A, so that α β =, α + β = A, αβ = B hence by (1) () U n = αn β n Lemma 1. For any integer n 0 we have U 3n = 3U n B n + U 3 n. Proof. By (), using that αβ = B, we have to prove that α 3n β 3n = 3 αn β n ( α (αβ) n n β n ) 3 +,

3 Some congruences concerning second order linear recurrences 31 which follows from α 3n β 3n = 3(α n β n )α n β n + (α n β n ) 3. Lemma. For any non-negative integers m n we have U m+n = V n U m+n B n U m. Proof. Similarly as in the proof of Lemma 1, α m+n β m+n = (α n + β n ) αm+n β m+n (αβ) n αm β m is an identity which by (1) (), implies the lemma. Lemma 3. For any n 0 we have V n = B n + U n = V n B n U n = U n V n. Proof. The identities ( α α n +β n = (αβ) n n β n ) + αn β n = αn β n (α n + β n ) prove the lemma. Proof of the Theorem. We prove the first congruence of the Theorem by double induction on k. For k = 1 k = 3, by Lemma 1, the congruence is an identity. Suppose the congruence holds for k k +, where k 1 is odd. Then by Lemma 3 we have (3) U n(k+4) = U nk+4n = V n U nk+n B n U nk = (B n + U n)u n(k+) B n U nk (B n + U n )Q Bn R (mod U 5 n ), where (4) Q = (k + )B k+1 n U n + (k + )((k + ) 1) B k 1 n Un 3 (5) R = kb k 1 n U n + k(k 1) B k 3 n U 3 n.

4 3 James P. Jones Péter Kiss After some calculation (3), (4) (5) imply (6) U n(k+4) U n T + U 3 n S (mod U 5 n ), where T = ((k + ) k)b k+3 n = (k + 4)B (k+4) 1 S = (k + )B k+1 n + (k + )( (k + ) 1 ) k(k 1) B k+1 n = (k + 4)( (k + 4) 1 ) so by (6), U n(k+4) (k + 4)B (k+4) 1 U n + (k + 4)( (k + 4) 1 ) B k+1 n B (k+4) 3 n, B (k+4) 3 n U 3 n (mod U 5 n ). Hence the congruence holds also for k + 4 for any odd positive integer k. The other congruences in the Theorem can be proved similarly using Lemma 1,, 3 the identities U n = V n U n, V n = V n B n = B n + U n, U 3n = U n V n B n U n, V 3n = V 3 n 3Bn V n = B n V n + V n U n, U 4n = U n V 3 n B n U n V n, V 4n = V 4 n 4B n V n + B n. References [1]. Jarden, Recurring sequences, Riveon Lematematika, Jerusalem (Israel), [] J. P. Jones P. Kiss, Some identities congruences for a special family of second order recurrences, Acta Acad. Paed. Agriensis, Sect. Math. 3 ( ), 3 9.

5 Some congruences concerning second order linear recurrences 33 [3] J. P. Jones P. Kiss, Some new identities congruences for Lucas sequences, iscuss Math., to appear. [4]. H. Lehmer, On the multiple solutions of the Pell Equation, Annals of Math. 30 (198), [5]. H. Lehmer, An extended theory of Lucas functions, Annals of Math. 31 (1930), [6] E. Lucas, Theorie des functions numériques simplement périodiques, American Journal of Mathematics, vol. 1 (1878), 184 0, English translation: Fibonacci Association, Santa Clara Univ., [7] S. Vajda, Fibonacci & Lucas numbers, the golden section, Ellis Horwood Limited Publ., New York-Toronto, [8] C. R. Wall, Some congruences involving generalized Fibonacci numbers, The Fibonacci Quarterly 17.1 (1979), James P. Jones epartment of Mathematics Statistics University of Calgary Calgary, Alberta TN 1N4 Canada Péter Kiss Eszterházy Károly Teachers Training College epartment of Mathematics Leányka u Eger, Pf. 43. Hungary kissp@gemini.ektf.hu

In memoriam Péter Kiss

In memoriam Péter Kiss Kálmán Liptai Eszterházy Károly Collage, Leányka út 4, 3300 Eger, Hungary e-mail: liptaik@gemini.ektf.hu (Submitted March 2002-Final Revision October 2003) In memoriam Péter Kiss ABSTRACT A positive integer

More information

ON GENERALIZED BALANCING SEQUENCES

ON GENERALIZED BALANCING SEQUENCES ON GENERALIZED BALANCING SEQUENCES ATTILA BÉRCZES, KÁLMÁN LIPTAI, AND ISTVÁN PINK Abstract. Let R i = R(A, B, R 0, R 1 ) be a second order linear recurrence sequence. In the present paper we prove that

More information

Further generalizations of the Fibonacci-coefficient polynomials

Further generalizations of the Fibonacci-coefficient polynomials Annales Mathematicae et Informaticae 35 (2008) pp 123 128 http://wwwektfhu/ami Further generalizations of the Fibonacci-coefficient polynomials Ferenc Mátyás Institute of Mathematics and Informatics Eszterházy

More information

Acta Acad. Paed. Agriensis, Sectio Mathematicae 28 (2001) THE LIE AUGMENTATION TERMINALS OF GROUPS. Bertalan Király (Eger, Hungary)

Acta Acad. Paed. Agriensis, Sectio Mathematicae 28 (2001) THE LIE AUGMENTATION TERMINALS OF GROUPS. Bertalan Király (Eger, Hungary) Acta Acad. Paed. Agriensis, Sectio Mathematicae 28 (2001) 93 97 THE LIE AUGMENTATION TERMINALS OF GROUPS Bertalan Király (Eger, Hungary) Abstract. In this paper we give necessary and sufficient conditions

More information

THE p-adic VALUATION OF LUCAS SEQUENCES

THE p-adic VALUATION OF LUCAS SEQUENCES THE p-adic 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 information

A characterization of the identity function

A characterization of the identity function Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae, 4. 1997) pp. 3 9 A characterization of the identity function BUI MINH PHONG Abstract. We prove that if a multiplicative function f satisfies

More information

GENERALIZED FIBONACCI AND LUCAS NUMBERS OF THE FORM wx 2 AND wx 2 1

GENERALIZED 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

198 VOLUME 46/47, NUMBER 3

198 VOLUME 46/47, NUMBER 3 LAWRENCE SOMER Abstract. Rotkiewicz has shown that there exist Fibonacci pseudoprimes having the forms p(p + 2), p(2p 1), and p(2p + 3), where all the terms in the products are odd primes. Assuming Dickson

More information

Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 30 (2003) ON ADDITIVE FUNCTIONS SATISFYING CONGRUENCE PROPERTIES

Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 30 (2003) ON ADDITIVE FUNCTIONS SATISFYING CONGRUENCE PROPERTIES Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 30 2003) 123 132 ON ADDITIVE FUNCTIONS SATISFYING CONGRUENCE PROPERTIES Bui Minh Phong Budapest, Hungary) Dedicated to the memory of Professor

More information

Introduction to Lucas Sequences

Introduction to Lucas Sequences A talk given at Liaoning Normal Univ. (Dec. 14, 017) Introduction to Lucas Sequences Zhi-Wei Sun Nanjing University Nanjing 10093, P. R. China zwsun@nju.edu.cn http://math.nju.edu.cn/ zwsun Dec. 14, 017

More information

Acta Acad. Paed. Agriensis, Sectio Mathematicae 28 (2001) 21 26

Acta Acad. Paed. Agriensis, Sectio Mathematicae 28 (2001) 21 26 Acta Acad. Paed. Agriensis, Sectio Mathematicae 8 (00) 6 APPROXIMATION BY QUOTIENTS OF TERMS OF SECOND ORDER LINEAR RECURSIVE SEQUENCES OF INTEGERS Sándor H.-Molnár (Budapest, Hungary) Abstract. In the

More information

Mathematica Slovaca. Jaroslav Hančl; Péter Kiss On reciprocal sums of terms of linear recurrences. Terms of use:

Mathematica Slovaca. Jaroslav Hančl; Péter Kiss On reciprocal sums of terms of linear recurrences. Terms of use: Mathematica Slovaca Jaroslav Hančl; Péter Kiss On reciprocal sums of terms of linear recurrences Mathematica Slovaca, Vol. 43 (1993), No. 1, 31--37 Persistent URL: http://dml.cz/dmlcz/136572 Terms of use:

More information

G. Sburlati GENERALIZED FIBONACCI SEQUENCES AND LINEAR RECURRENCES

G. Sburlati GENERALIZED FIBONACCI SEQUENCES AND LINEAR RECURRENCES Rend. Sem. Mat. Univ. Pol. Torino - Vol. 65, 3 (2007) G. Sburlati GENERALIZED FIBONACCI SEQUENCES AND LINEAR RECURRENCES Abstract. We analyze the existing relations among particular classes of generalized

More information

Acta Acad. Paed. Agriensis, Sectio Mathematicae 28 (2001) POWER INTEGRAL BASES IN MIXED BIQUADRATIC NUMBER FIELDS

Acta Acad. Paed. Agriensis, Sectio Mathematicae 28 (2001) POWER INTEGRAL BASES IN MIXED BIQUADRATIC NUMBER FIELDS Acta Acad Paed Agriensis, Sectio Mathematicae 8 00) 79 86 POWER INTEGRAL BASES IN MIXED BIQUADRATIC NUMBER FIELDS Gábor Nyul Debrecen, Hungary) Abstract We give a complete characterization of power integral

More information

DISTRIBUTION OF FIBONACCI AND LUCAS NUMBERS MODULO 3 k

DISTRIBUTION OF FIBONACCI AND LUCAS NUMBERS MODULO 3 k DISTRIBUTION OF FIBONACCI AND LUCAS NUMBERS MODULO 3 k RALF BUNDSCHUH AND PETER BUNDSCHUH Dedicated to Peter Shiue on the occasion of his 70th birthday Abstract. Let F 0 = 0,F 1 = 1, and F n = F n 1 +F

More information

Diophantine triples in a Lucas-Lehmer sequence

Diophantine triples in a Lucas-Lehmer sequence Annales Mathematicae et Informaticae 49 (01) pp. 5 100 doi: 10.33039/ami.01.0.001 http://ami.uni-eszterhazy.hu Diophantine triples in a Lucas-Lehmer sequence Krisztián Gueth Lorand Eötvös University Savaria

More information

#A11 INTEGERS 12 (2012) FIBONACCI VARIATIONS OF A CONJECTURE OF POLIGNAC

#A11 INTEGERS 12 (2012) FIBONACCI VARIATIONS OF A CONJECTURE OF POLIGNAC #A11 INTEGERS 12 (2012) FIBONACCI VARIATIONS OF A CONJECTURE OF POLIGNAC Lenny Jones Department of Mathematics, Shippensburg University, Shippensburg, Pennsylvania lkjone@ship.edu Received: 9/17/10, Revised:

More information

Chris Smyth School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3JZ, UK.

Chris Smyth School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3JZ, UK. THE DIVISIBILITY OF a n b n BY POWERS OF n Kálmán Győry Institute of Mathematics, University of Debrecen, Debrecen H-4010, Hungary. gyory@math.klte.hu Chris Smyth School of Mathematics and Maxwell Institute

More information

ALTERNATING SUMS OF FIBONACCI PRODUCTS

ALTERNATING 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 information

Burkholder s inequality for multiindex martingales

Burkholder s inequality for multiindex martingales Annales Mathematicae et Informaticae 32 (25) pp. 45 51. Burkholder s inequality for multiindex martingales István Fazekas Faculty of Informatics, University of Debrecen e-mail: fazekasi@inf.unideb.hu Dedicated

More information

SOME SUMS FORMULAE FOR PRODUCTS OF TERMS OF PELL, PELL- LUCAS AND MODIFIED PELL SEQUENCES

SOME SUMS FORMULAE FOR PRODUCTS OF TERMS OF PELL, PELL- LUCAS AND MODIFIED PELL SEQUENCES SOME SUMS FORMULAE FOR PRODUCTS OF TERMS OF PELL PELL- LUCAS AND MODIFIED PELL SEQUENCES Serpil HALICI Sakarya Üni. Sciences and Arts Faculty Dept. of Math. Esentepe Campus Sakarya. shalici@ssakarya.edu.tr

More information

On the Shifted Product of Binary Recurrences

On the Shifted Product of Binary Recurrences 1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 13 (2010), rticle 10.6.1 On the Shifted Product of Binary Recurrences Omar Khadir epartment of Mathematics University of Hassan II Mohammedia, Morocco

More information

When do Fibonacci invertible classes modulo M form a subgroup?

When 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 information

Florian Luca Instituto de Matemáticas, Universidad Nacional Autonoma de México, C.P , Morelia, Michoacán, México

Florian Luca Instituto de Matemáticas, Universidad Nacional Autonoma de México, C.P , Morelia, Michoacán, México Florian Luca Instituto de Matemáticas, Universidad Nacional Autonoma de México, C.P. 8180, Morelia, Michoacán, México e-mail: fluca@matmor.unam.mx Laszlo Szalay Department of Mathematics and Statistics,

More information

QUOTIENTS OF FIBONACCI NUMBERS

QUOTIENTS OF FIBONACCI NUMBERS QUOTIENTS OF FIBONACCI NUMBERS STEPHAN RAMON GARCIA AND FLORIAN LUCA Abstract. There have been many articles in the Monthly on quotient sets over the years. We take a first step here into the p-adic setting,

More information

CALCULATING EXACT CYCLE LENGTHS IN THE GENERALIZED FIBONACCI SEQUENCE MODULO p

CALCULATING EXACT CYCLE LENGTHS IN THE GENERALIZED FIBONACCI SEQUENCE MODULO p CALCULATING EXACT CYCLE LENGTHS IN THE GENERALIZED FIBONACCI SEQUENCE MODULO p DOMINIC VELLA AND ALFRED VELLA. Introduction The cycles that occur in the Fibonacci sequence {F n } n=0 when it is reduced

More information

Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004) 19 24

Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004) 19 24 Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004) 19 24 PRIMITIVE DIVISORS OF LUCAS SEQUENCES AND PRIME FACTORS OF x 2 + 1 AND x 4 + 1 Florian Luca (Michoacán, México) Abstract. In this

More information

Arithmetic properties of lacunary sums of binomial coefficients

Arithmetic properties of lacunary sums of binomial coefficients Arithmetic properties of lacunary sums of binomial coefficients Tamás Mathematics Department Occidental College 29th Journées Arithmétiques JA2015, July 6-10, 2015 Arithmetic properties of lacunary sums

More information

1. Introduction. Let P and Q be non-zero relatively prime integers, α and β (α > β) be the zeros of x 2 P x + Q, and, for n 0, let

1. Introduction. Let P and Q be non-zero relatively prime integers, α and β (α > β) be the zeros of x 2 P x + Q, and, for n 0, let C O L L O Q U I U M M A T H E M A T I C U M VOL. 78 1998 NO. 1 SQUARES IN LUCAS SEQUENCES HAVING AN EVEN FIRST PARAMETER BY PAULO R I B E N B O I M (KINGSTON, ONTARIO) AND WAYNE L. M c D A N I E L (ST.

More information

Divisibility in the Fibonacci Numbers. Stefan Erickson Colorado College January 27, 2006

Divisibility in the Fibonacci Numbers. Stefan Erickson Colorado College January 27, 2006 Divisibility in the Fibonacci Numbers Stefan Erickson Colorado College January 27, 2006 Fibonacci Numbers F n+2 = F n+1 + F n n 1 2 3 4 6 7 8 9 10 11 12 F n 1 1 2 3 8 13 21 34 89 144 n 13 14 1 16 17 18

More information

#A5 INTEGERS 17 (2017) THE 2-ADIC ORDER OF SOME GENERALIZED FIBONACCI NUMBERS

#A5 INTEGERS 17 (2017) THE 2-ADIC ORDER OF SOME GENERALIZED FIBONACCI NUMBERS #A5 INTEGERS 7 (207) THE 2-ADIC ORDER OF SOME GENERALIZED FIBONACCI NUMBERS Tamás Lengyel Mathematics Department, Occidental College, Los Angeles, California lengyel@oxy.edu Diego Marques Departamento

More information

Summation of Certain Infinite Lucas-Related Series

Summation of Certain Infinite Lucas-Related Series J. Integer Sequences 22 (209) Article 9..6. Summation of Certain Infinite Lucas-Related Series arxiv:90.04336v [math.nt] Jan 209 Bakir Farhi Laboratoire de Mathématiques appliquées Faculté des Sciences

More information

Some sophisticated congruences involving Fibonacci numbers

Some sophisticated congruences involving Fibonacci numbers A tal given at the National Center for Theoretical Sciences (Hsinchu, Taiwan; July 20, 2011 and Shanghai Jiaotong University (Nov. 4, 2011 Some sohisticated congruences involving Fibonacci numbers Zhi-Wei

More information

Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 30 (2003) A NOTE ON NON-NEGATIVE INFORMATION FUNCTIONS

Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 30 (2003) A NOTE ON NON-NEGATIVE INFORMATION FUNCTIONS Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 30 (2003 3 36 A NOTE ON NON-NEGATIVE INFORMATION FUNCTIONS Béla Brindza and Gyula Maksa (Debrecen, Hungary Dedicated to the memory of Professor

More information

arxiv: v1 [math.nt] 2 Oct 2015

arxiv: v1 [math.nt] 2 Oct 2015 PELL NUMBERS WITH LEHMER PROPERTY arxiv:1510.00638v1 [math.nt] 2 Oct 2015 BERNADETTE FAYE FLORIAN LUCA Abstract. In this paper, we prove that there is no number with the Lehmer property in the sequence

More information

On second order non-homogeneous recurrence relation

On second order non-homogeneous 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 information

A note on the products of the terms of linear recurrences

A note on the products of the terms of linear recurrences Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae, 24 (1997) pp 47 53 A note on the products of the terms of linear recurrences LÁSZLÓ SZALAY Abstract For an integer ν>1 let G (i) (,,ν) be linear

More information

On Sums of Products of Horadam Numbers

On Sums of Products of Horadam Numbers KYUNGPOOK Math J 49(009), 483-49 On Sums of Products of Horadam Numbers Zvonko ƒerin Kopernikova 7, 10010 Zagreb, Croatia, Europe e-mail : cerin@mathhr Abstract In this paper we give formulae for sums

More information

arxiv: v1 [math.nt] 19 Dec 2018

arxiv: v1 [math.nt] 19 Dec 2018 ON SECOND ORDER LINEAR SEQUENCES OF COMPOSITE NUMBERS DAN ISMAILESCU 2, ADRIENNE KO 1, CELINE LEE 3, AND JAE YONG PARK 4 arxiv:1812.08041v1 [math.nt] 19 Dec 2018 Abstract. In this paper we present a new

More information

NOTE ON FIBONACCI PRIMALITY TESTING John Brillhart University of Arizona, Tucson, AZ (Submitted August 1996-Final Revision January 1997)

NOTE ON FIBONACCI PRIMALITY TESTING John Brillhart University of Arizona, Tucson, AZ (Submitted August 1996-Final Revision January 1997) John Brillhart University of Arizona, Tucson, AZ 85721 (Submitted August 1996-Final Revision January 1997) 1. INTRODUCTION One of the most effective ways of proving an integer N is prime is to show first

More information

On repdigits as product of consecutive Lucas numbers

On repdigits as product of consecutive Lucas numbers Notes on Number Theory and Discrete Mathematics Print ISSN 1310 5132, Online ISSN 2367 8275 Vol. 24, 2018, No. 3, 5 102 DOI: 10.7546/nntdm.2018.24.3.5-102 On repdigits as product of consecutive Lucas numbers

More information

A CRITERION FOR POLYNOMIALS TO BE CONGRUENT TO THE PRODUCT OF LINEAR POLYNOMIALS (mod p) ZHI-HONG SUN

A CRITERION FOR POLYNOMIALS TO BE CONGRUENT TO THE PRODUCT OF LINEAR POLYNOMIALS (mod p) ZHI-HONG SUN A CRITERION FOR POLYNOMIALS TO BE CONGRUENT TO THE PRODUCT OF LINEAR POLYNOMIALS (mod ) ZHI-HONG SUN Deartment of Mathematics, Huaiyin Teachers College, Huaian 223001, Jiangsu, P. R. China e-mail: hyzhsun@ublic.hy.js.cn

More information

Even perfect numbers among generalized Fibonacci sequences

Even perfect numbers among generalized Fibonacci sequences Even perfect numbers among generalized Fibonacci sequences Diego Marques 1, Vinicius Vieira 2 Departamento de Matemática, Universidade de Brasília, Brasília, 70910-900, Brazil Abstract A positive integer

More information

THE ORDER OF APPEARANCE OF PRODUCT OF CONSECUTIVE FIBONACCI NUMBERS

THE ORDER OF APPEARANCE OF PRODUCT OF CONSECUTIVE FIBONACCI NUMBERS THE ORDER OF APPEARANCE OF PRODUCT OF CONSECUTIVE FIBONACCI NUMBERS DIEGO MARQUES Abstract. Let F n be the nth Fibonacci number. The order of appearance z(n) of a natural number n is defined as the smallest

More information

Generalized Lucas Sequences Part II

Generalized Lucas Sequences Part II Introduction Generalized Lucas Sequences Part II Daryl DeFord Washington State University February 4, 2013 Introduction Èdouard Lucas: The theory of recurrent sequences is an inexhaustible mine which contains

More information

On k-fibonacci Numbers with Applications to Continued Fractions

On k-fibonacci Numbers with Applications to Continued Fractions Journal of Physics: Conference Series PAPER OPEN ACCESS On k-fibonacci Numbers with Applications to Continued Fractions Related content - Some results on circulant and skew circulant type matrices with

More information

On Generalized k-fibonacci Sequence by Two-Cross-Two Matrix

On Generalized k-fibonacci Sequence by Two-Cross-Two 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 k-fibonacci

More information

Cullen Numbers in Binary Recurrent Sequences

Cullen Numbers in Binary Recurrent Sequences Cullen Numbers in Binary Recurrent Sequences Florian Luca 1 and Pantelimon Stănică 2 1 IMATE-UNAM, Ap. Postal 61-3 (Xangari), CP 58 089 Morelia, Michoacán, Mexico; e-mail: fluca@matmor.unam.mx 2 Auburn

More information

A PROOF OF MELHAM S CONJECTURE

A PROOF OF MELHAM S CONJECTURE A PROOF OF MELHAM S CONJECTRE EMRAH KILIC 1, ILKER AKKS, AND HELMT PRODINGER 3 Abstract. In this paper, we consider Melha s conecture involving Fibonacci and Lucas nubers. After rewriting it in ters of

More information

ON SUMS OF SQUARES OF PELL-LUCAS NUMBERS. Gian Mario Gianella University of Torino, Torino, Italy, Europe.

ON SUMS OF SQUARES OF PELL-LUCAS 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 PELL-LUCAS NUMBERS Zvonko Čerin University of Zagreb, Zagreb, Croatia, Europe cerin@math.hr Gian Mario Gianella

More information

Diophantine m-tuples and elliptic curves

Diophantine m-tuples and elliptic curves Diophantine m-tuples and elliptic curves Andrej Dujella (Zagreb) 1 Introduction Diophantus found four positive rational numbers 1 16, 33 16, 17 4, 105 16 with the property that the product of any two of

More information

When do the Fibonacci invertible classes modulo M form a subgroup?

When do the Fibonacci invertible classes modulo M form a subgroup? Annales Mathematicae et Informaticae 41 (2013). 265 270 Proceedings of the 15 th International Conference on Fibonacci Numbers and Their Alications Institute of Mathematics and Informatics, Eszterházy

More information

ON EXPLICIT FORMULAE AND LINEAR RECURRENT SEQUENCES. 1. Introduction

ON EXPLICIT FORMULAE AND LINEAR RECURRENT SEQUENCES. 1. Introduction ON EXPLICIT FORMULAE AND LINEAR RECURRENT SEQUENCES R. EULER and L. H. GALLARDO Abstract. We notice that some recent explicit results about linear recurrent sequences over a ring R with 1 were already

More information

Notes on Continued Fractions for Math 4400

Notes on Continued Fractions for Math 4400 . Continued fractions. Notes on Continued Fractions for Math 4400 The continued fraction expansion converts a positive real number α into a sequence of natural numbers. Conversely, a sequence of natural

More information

A Variation of a Congruence of Subbarao for n = 2 α 5 β, α 0, β 0

A Variation of a Congruence of Subbarao for n = 2 α 5 β, α 0, β 0 Introduction A Variation of a Congruence of Subbarao for n = 2 α 5 β, α 0, β 0 Diophantine Approximation and Related Topics, 2015 Aarhus, Denmark Sanda Bujačić 1 1 Department of Mathematics University

More information

On arithmetic functions of balancing and Lucas-balancing numbers

On arithmetic functions of balancing and Lucas-balancing numbers MATHEMATICAL COMMUNICATIONS 77 Math. Commun. 24(2019), 77 1 On arithmetic functions of balancing and Lucas-balancing numbers Utkal Keshari Dutta and Prasanta Kumar Ray Department of Mathematics, Sambalpur

More information

MATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences.

MATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences. MATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences. Congruences Let n be a postive integer. The integers a and b are called congruent modulo n if they have the same

More information

Impulse Response Sequences and Construction of Number Sequence Identities

Impulse Response Sequences and Construction of Number Sequence Identities Impulse Response Sequences and Construction of Number Sequence Identities Tian-Xiao He Department of Mathematics Illinois Wesleyan University Bloomington, IL 6170-900, USA Abstract As an extension of Lucas

More information

The Fibonacci Quarterly 44(2006), no.2, PRIMALITY TESTS FOR NUMBERS OF THE FORM k 2 m ± 1. Zhi-Hong Sun

The Fibonacci Quarterly 44(2006), no.2, PRIMALITY TESTS FOR NUMBERS OF THE FORM k 2 m ± 1. Zhi-Hong Sun The Fibonacci Quarterly 44006, no., 11-130. PRIMALITY TESTS FOR NUMBERS OF THE FORM k m ± 1 Zhi-Hong Sun eartment of Mathematics, Huaiyin Teachers College, Huaian, Jiangsu 3001, P.R. China E-mail: zhsun@hytc.edu.cn

More information

ELEMENTARY PROBLEMS AND SOLUTIONS. Edited by Stanley Rabinowitz

ELEMENTARY PROBLEMS AND SOLUTIONS. Edited by Stanley Rabinowitz Edited by Stanley Rabinowitz Please send all material for to Dr. STANLEY RABINOWITZ; 12 VINE BROOK RD; WESTFORD f AM 01886-4212 USA. Correspondence may also be sent to the problem editor by electronic

More information

MATRICES AND LINEAR RECURRENCES IN FINITE FIELDS

MATRICES AND LINEAR RECURRENCES IN FINITE FIELDS Owen J. Brison Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, Bloco C6, Piso 2, Campo Grande, 1749-016 LISBOA, PORTUGAL e-mail: brison@ptmat.fc.ul.pt J. Eurico Nogueira Departamento

More information

Certain Diophantine equations involving balancing and Lucas-balancing numbers

Certain Diophantine equations involving balancing and Lucas-balancing numbers ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 0, Number, December 016 Available online at http://acutm.math.ut.ee Certain Diophantine equations involving balancing and Lucas-balancing

More information

SECOND-ORDER RECURRENCES. Lawrence Somer Department of Mathematics, Catholic University of America, Washington, D.C

SECOND-ORDER RECURRENCES. Lawrence Somer Department of Mathematics, Catholic University of America, Washington, D.C p-stability OF DEGENERATE SECOND-ORDER RECURRENCES Lawrence Somer Department of Mathematics, Catholic University of America, Washington, D.C. 20064 Walter Carlip Department of Mathematics and Computer

More information

Fifteen problems in number theory

Fifteen problems in number theory Acta Univ. Sapientiae, Mathematica, 2, 1 (2010) 72 83 Fifteen problems in number theory Attila Pethő University of Debrecen Department of Computer Science and Number Theory Research Group Hungarian Academy

More information

Journal of Number Theory

Journal of Number Theory Journal of Number Theory 130 (2010) 1737 1749 Contents lists available at ScienceDirect Journal of Number Theory www.elsevier.com/locate/jnt A binary linear recurrence sequence of composite numbers Artūras

More information

A PROOF OF A CONJECTURE OF MELHAM

A 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 information

N O N E X I S T E N C E O F EVEN FIBONACCI P S E U D O P R I M E S O F THE P T KIND*

N O N E X I S T E N C E O F EVEN FIBONACCI P S E U D O P R I M E S O F THE P T KIND* N O N E X I S T E N C E O F EVEN FIBONACCI P S E U D O P R I M E S O F THE P T KIND* Adina DI Porto Fondazione Ugo Bordoni, Rome, Italy (Submitted August 1991) 1. INTRODUCTION AND PRELIMINARIES Fibonacci

More information

On the spacings between C-nomial coefficients

On the spacings between C-nomial coefficients On the spacings between C-nomial coefficients lorian Luca, Diego Marques,2, Pantelimon Stănică 3 Instituto de Matemáticas, Universidad Nacional Autónoma de México C.P. 58089, Morelia, Michoacán, México,

More information

RICCATI MEETS FIBONACCI

RICCATI MEETS FIBONACCI Wolfdieter Lang Institut für Theoretische Physik, Universität Karlsruhe Kaiserstrasse 2, D-763 Karlsruhe, Germany e-mail: wolfdieter.lang@physik.uni-karlsruhe.de (Submitted November 200- Final Revision

More information

Formulas for sums of squares and products of Pell numbers

Formulas for sums of squares and products of Pell numbers Formulas for sums of squares and products of Pell numbers TEORIA DEI NUMERI Nota di Zvonko ČERIN e Gian Mario GIANELLA presentata dal socio Corrispondente Marius I. STOKA nell adunanza del 1 Giugno 2006.

More information

arxiv: v2 [math.nt] 29 Jul 2017

arxiv: v2 [math.nt] 29 Jul 2017 Fibonacci and Lucas Numbers Associated with Brocard-Ramanujan Equation arxiv:1509.07898v2 [math.nt] 29 Jul 2017 Prapanpong Pongsriiam Department of Mathematics, Faculty of Science Silpakorn University

More information

1 Examples of Weak Induction

1 Examples of Weak Induction More About Mathematical Induction Mathematical induction is designed for proving that a statement holds for all nonnegative integers (or integers beyond an initial one). Here are some extra examples of

More information

ABSTRACT 1. INTRODUCTION

ABSTRACT 1. INTRODUCTION ON PIMES AN TEMS OF PIME O INEX IN THE LEHME SEUENCES John H. Jaroma epartment of Mathematical Sciences, Loyola College in Maryland, Baltimore, M 110 Submitted May 004-Final evision March 006 ABSTACT It

More information

CONGRUENCES FOR BERNOULLI - LUCAS SUMS

CONGRUENCES FOR BERNOULLI - LUCAS SUMS CONGRUENCES FOR BERNOULLI - LUCAS SUMS PAUL THOMAS YOUNG Abstract. We give strong congruences for sums of the form N BnVn+1 where Bn denotes the Bernoulli number and V n denotes a Lucas sequence of the

More information

Problem Set 5 Solutions

Problem Set 5 Solutions Problem Set 5 Solutions Section 4.. Use mathematical induction to prove each of the following: a) For each natural number n with n, n > + n. Let P n) be the statement n > + n. The base case, P ), is true

More information

Math 324 Summer 2012 Elementary Number Theory Notes on Mathematical Induction

Math 324 Summer 2012 Elementary Number Theory Notes on Mathematical Induction Math 4 Summer 01 Elementary Number Theory Notes on Mathematical Induction Principle of Mathematical Induction Recall the following axiom for the set of integers. Well-Ordering Axiom for the Integers If

More information

PODSYPANIN S PAPER ON THE LENGTH OF THE PERIOD OF A QUADRATIC IRRATIONAL. Copyright 2007

PODSYPANIN S PAPER ON THE LENGTH OF THE PERIOD OF A QUADRATIC IRRATIONAL. Copyright 2007 PODSYPANIN S PAPER ON THE LENGTH OF THE PERIOD OF A QUADRATIC IRRATIONAL JOHN ROBERTSON Copyright 007 1. Introduction The major result in Podsypanin s paper Length of the period of a quadratic irrational

More information

NEW IDENTITIES FOR THE COMMON FACTORS OF BALANCING AND LUCAS-BALANCING NUMBERS

NEW IDENTITIES FOR THE COMMON FACTORS OF BALANCING AND LUCAS-BALANCING NUMBERS International Journal of Pure and Applied Mathematics Volume 85 No. 3 013, 487-494 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.173/ijpam.v85i3.5

More information

AND RELATED NUMBERS. GERHARD ROSENBERGER Dortmund, Federal Republic of Germany (Submitted April 1982)

AND RELATED NUMBERS. GERHARD ROSENBERGER Dortmund, Federal Republic of Germany (Submitted April 1982) ON SOME D I V I S I B I L I T Y PROPERTIES OF FIBONACCI AND RELATED NUMBERS Universitat GERHARD ROSENBERGER Dortmund, Federal Republic of Germany (Submitted April 1982) 1. Let x be an arbitrary natural

More information

ESCALATOR NUMBER SEQUENCES

ESCALATOR NUMBER SEQUENCES H. G. GRUNDMAN Abstract. An escalator sequence is an infinite sequence of rational numbers such that each partial sum is equal to the corresponding partial product. An escalator number is one of these

More information

FIFTH ROOTS OF FIBONACCI FRACTIONS. Christopher P. French Grinnell College, Grinnell, IA (Submitted June 2004-Final Revision September 2004)

FIFTH ROOTS OF FIBONACCI FRACTIONS. Christopher P. French Grinnell College, Grinnell, IA (Submitted June 2004-Final Revision September 2004) Christopher P. French Grinnell College, Grinnell, IA 0112 (Submitted June 2004-Final Revision September 2004) ABSTRACT We prove that when n is odd, the continued fraction expansion of Fn+ begins with a

More information

Fall 2017 Test II review problems

Fall 2017 Test II review problems Fall 2017 Test II review problems Dr. Holmes October 18, 2017 This is a quite miscellaneous grab bag of relevant problems from old tests. Some are certainly repeated. 1. Give the complete addition and

More information

SOME SUMMATION IDENTITIES USING GENERALIZED g-matrices

SOME SUMMATION IDENTITIES USING GENERALIZED g-matrices SOME SUMMATION IDENTITIES USING GENERALIZED g-matrices R. S. Melham and A. G. Shannon University of Technology, Sydney, 2007, Australia {Submitted June 1993) 1. INTRODUCTION In a belated acknowledgment,

More information

#A61 INTEGERS 12 (2012) ON FINITE SUMS OF GOOD AND SHAR THAT INVOLVE RECIPROCALS OF FIBONACCI NUMBERS

#A61 INTEGERS 12 (2012) ON FINITE SUMS OF GOOD AND SHAR THAT INVOLVE RECIPROCALS OF FIBONACCI NUMBERS #A6 INTEGERS 2 (202) ON INITE SUMS O GOOD AND SHAR THAT INVOLVE RECIPROCALS O IBONACCI NUMBERS R. S. Melham School of Mathematical Sciences, University of Technology, Sydney, Australia ray.melham@uts.edu.au

More information

On Some Combinations of Non-Consecutive Terms of a Recurrence Sequence

On Some Combinations of Non-Consecutive Terms of a Recurrence Sequence 1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 21 (2018), Article 18.3.5 On Some Combinations of Non-Consecutive Terms of a Recurrence Sequence Eva Trojovská Department of Mathematics Faculty of Science

More information

4 Linear Recurrence Relations & the Fibonacci Sequence

4 Linear Recurrence Relations & the Fibonacci Sequence 4 Linear Recurrence Relations & the Fibonacci Sequence Recall the classic example of the Fibonacci sequence (F n ) n=1 the relations: F n+ = F n+1 + F n F 1 = F = 1 = (1, 1,, 3, 5, 8, 13, 1,...), defined

More information

arxiv: v1 [math.co] 11 Aug 2015

arxiv: v1 [math.co] 11 Aug 2015 arxiv:1508.02762v1 [math.co] 11 Aug 2015 A Family of the Zeckendorf Theorem Related Identities Ivica Martinjak Faculty of Science, University of Zagreb Bijenička cesta 32, HR-10000 Zagreb, Croatia Abstract

More information

PELL NUMBERS WHOSE EULER FUNCTION IS A PELL NUMBER. Bernadette Faye and Florian Luca

PELL NUMBERS WHOSE EULER FUNCTION IS A PELL NUMBER. Bernadette Faye and Florian Luca PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle série tome 05 207 23 245 https://doiorg/02298/pim7523f PELL NUMBERS WHOSE EULER FUNCTION IS A PELL NUMBER Bernadette Faye and Florian Luca Abstract We show

More information

Fermat s Little Theorem. Fermat s little theorem is a statement about primes that nearly characterizes them.

Fermat s Little Theorem. Fermat s little theorem is a statement about primes that nearly characterizes them. Fermat s Little Theorem Fermat s little theorem is a statement about primes that nearly characterizes them. Theorem: Let p be prime and a be an integer that is not a multiple of p. Then a p 1 1 (mod p).

More information

^ ) = ^ : g ' f ^4(x)=a(xr + /xxr,

^ ) = ^ : g ' f ^4(x)=a(xr + /xxr, Edited by Stanley Rablnowitz Please send all material for ELEMENTARY PROBLEMS AND SOLUTIONS to Dr. STANLEY RABINOWITZ; 12 VINE BROOK RD; WESTFORD, MA 01886-4212 USA. Correspondence may also be sent to

More information

Gaussian Modified Pell Sequence and Gaussian Modified Pell Polynomial Sequence

Gaussian Modified Pell Sequence and Gaussian Modified Pell Polynomial Sequence Aksaray University Journal of Science and Engineering e-issn: 2587-1277 http://dergipark.gov.tr/asujse http://asujse.aksaray.edu.tr Aksaray J. Sci. Eng. Volume 2, Issue 1, pp. 63-72 doi: 10.29002/asujse.374128

More information

Diophantine quadruples and Fibonacci numbers

Diophantine quadruples and Fibonacci numbers Diophantine quadruples and Fibonacci numbers Andrej Dujella Department of Mathematics, University of Zagreb, Croatia Abstract A Diophantine m-tuple is a set of m positive integers with the property that

More information

Prime Numbers in Generalized Pascal Triangles

Prime Numbers in Generalized Pascal Triangles Prime Numbers in Generalized Pascal Triangles G. Farkas, G. Kallós Eötvös Loránd University, H-1117, Budapest, Pázmány Péter sétány 1/C, farkasg@compalg.inf.elte.hu Széchenyi István University, H-9026,

More information

ON THE EXTENSIBILITY OF DIOPHANTINE TRIPLES {k 1, k + 1, 4k} FOR GAUSSIAN INTEGERS. Zrinka Franušić University of Zagreb, Croatia

ON 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 information

Notes on Systems of Linear Congruences

Notes on Systems of Linear Congruences MATH 324 Summer 2012 Elementary Number Theory Notes on Systems of Linear Congruences In this note we will discuss systems of linear congruences where the moduli are all different. Definition. Given the

More information

Generalized balancing numbers

Generalized balancing numbers Indag. Mathem., N.S., 20 (1, 87 100 March, 2009 Generalized balancing numbers by Kálmán Liptai a, Florian Luca b,ákospintér c and László Szalay d a Institute of Mathematics and Informatics, Eszterházy

More information

On the (s,t)-pell and (s,t)-pell-lucas numbers by matrix methods

On the (s,t)-pell and (s,t)-pell-lucas numbers by matrix methods Annales Mathematicae et Informaticae 46 06 pp 95 04 http://amiektfhu On the s,t-pell and s,t-pell-lucas numbers by matrix methods Somnuk Srisawat, Wanna Sriprad Department of Mathematics and computer science,

More information

PUTNAM PROBLEMS SEQUENCES, SERIES AND RECURRENCES. Notes

PUTNAM PROBLEMS SEQUENCES, SERIES AND RECURRENCES. Notes PUTNAM PROBLEMS SEQUENCES, SERIES AND RECURRENCES Notes. x n+ = ax n has the general solution x n = x a n. 2. x n+ = x n + b has the general solution x n = x + (n )b. 3. x n+ = ax n + b (with a ) can be

More information

Quadratic Diophantine Equations x 2 Dy 2 = c n

Quadratic Diophantine Equations x 2 Dy 2 = c n Irish Math. Soc. Bulletin 58 2006, 55 68 55 Quadratic Diophantine Equations x 2 Dy 2 c n RICHARD A. MOLLIN Abstract. We consider the Diophantine equation x 2 Dy 2 c n for non-square positive integers D

More information

A Remark on Prime Divisors of Lengths of Sides of Heron Triangles

A 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 information