Diophantine triples and reduced quadruples with the Lucas sequence of recurrence un=Aun-1-un-2

Nurettin Irmak;  ; László Szalay;

doi:10.3336/gm.49.2.05

Sebuah Generalisasi Baru Barisan Fibonacci-Lucas

Talenta Conference Series Science and Technology (ST) ◽

10.32734/st.v2i2.478 ◽

2019 ◽

Vol 2 (2) ◽

Author(s):

Musraini M Musraini M ◽

Rustam Efendi ◽

Rolan Pane ◽

Endang Lily

Keyword(s):

Recurrence Relation ◽

Initial Conditions ◽

Lucas Sequence ◽

Binet’S Formula

Barisan Fibonacci dan Lucas telah digeneralisasi dalam banyak cara, beberapa dengan mempertahankan kondisi awal, dan lainnya dengan mempertahankan relasi rekurensi. Makalah ini menyajikan sebuah generalisasi baru barisan Fibonacci-Lucas yang didefinisikan oleh relasi rekurensi B_n=B_(n-1)+B_(n-2),n≥2 , B_0=2b,B_1=s dengan b dan s bilangan bulat tak negatif. Selanjutnya, beberapa identitas dihasilkan dan diturunkan menggunakan formula Binet dan metode sederhana lainnya. Juga dibahas beberapa identitas dalam bentuk determinan. The Fibonacci and Lucas sequence has been generalized in many ways, some by preserving the initial conditions, and others by preserving the recurrence relation. In this paper, a new generalization of Fibonacci-Lucas sequence is introduced and defined by the recurrence relation B_n=B_(n-1)+B_(n-2),n≥2, with , B_0=2b,B_1=s where b and s are non negative integers. Further, some identities are generated and derived by Binet’s formula and other simple methods. Also some determinant identities are discussed.

Download Full-text

A CUBIC EL-GAMAL ENCRYPTION SCHEME BASED ON LUCAS SEQUENCE AND ELLIPTIC CURVE

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.10.11.5 ◽

2021 ◽

Vol 10 (11) ◽

pp. 3439-3447

Author(s):

T. J. Wong ◽

L. F. Koo ◽

F. H. Naning ◽

A. F. N. Rasedee ◽

M. M. Magiman ◽

...

Keyword(s):

Elliptic Curve ◽

Elliptic Curve Cryptography ◽

Mathematical Problem ◽

Public Key ◽

Public Key Cryptosystem ◽

Encryption Scheme ◽

Secret Key ◽

Chosen Plaintext Attack ◽

The Public ◽

Lucas Sequence

The public key cryptosystem is fundamental in safeguard communication in cyberspace. This paper described a new cryptosystem analogous to El-Gamal encryption scheme, which utilizing the Lucas sequence and Elliptic Curve. Similar to Elliptic Curve Cryptography (ECC) and Rivest-Shamir-Adleman (RSA), the proposed cryptosystem requires a precise hard mathematical problem as the essential part of security strength. The chosen plaintext attack (CPA) was employed to investigate the security of this cryptosystem. The result shows that the system is vulnerable against the CPA when the sender decrypts a plaintext with modified public key, where the cryptanalyst able to break the security of the proposed cryptosystem by recovering the plaintext even without knowing the secret key from either the sender or receiver.

Download Full-text

On the Sum of Powers of Two k-Fibonacci Numbers which Belongs to the Sequence of k-Lucas Numbers

Tatra Mountains Mathematical Publications ◽

10.1515/tmmp-2016-0028 ◽

2016 ◽

Vol 67 (1) ◽

pp. 41-46

Author(s):

Pavel Trojovský

Keyword(s):

Recurrence Relation ◽

Initial Conditions ◽

Fibonacci Numbers ◽

Fibonacci Sequence ◽

Lucas Numbers ◽

Initial Values ◽

Lucas Sequence ◽

Positive Integers

Abstract Let k ≥ 1 and denote (Fk,n)n≥0, the k-Fibonacci sequence whose terms satisfy the recurrence relation Fk,n = kFk,n−1 +Fk,n−2, with initial conditions Fk,0 = 0 and Fk,1 = 1. In the same way, the k-Lucas sequence (Lk,n)n≥0 is defined by satisfying the same recurrence relation with initial values Lk,0 = 2 and Lk,1 = k. These sequences were introduced by Falcon and Plaza, who showed many of their properties, too. In particular, they proved that Fk,n+1 + Fk,n−1 = Lk,n, for all k ≥ 1 and n ≥ 0. In this paper, we shall prove that if k ≥ 1 and $F_{k,n + 1}^s + F_{k,n - 1}^s \in \left( {L_{k,m} } \right)_{m \ge 1} $ for infinitely many positive integers n, then s =1.

Download Full-text

On the Lucas sequence equations Vn=kVmand Un=kUm

Colloquium Mathematicum ◽

10.4064/cm130-1-3 ◽

2013 ◽

Vol 130 (1) ◽

pp. 27-38 ◽

Cited By ~ 1

Author(s):

Refik Keskin ◽

Zafer Şiar

Keyword(s):

Lucas Sequence

Download Full-text

On some new results for the generalised Lucas sequences

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/auom-2021-0002 ◽

2021 ◽

Vol 29 (1) ◽

pp. 17-36

Author(s):

Dorin Andrica ◽

Ovidiu Bagdasar ◽

George Cătălin Ţurcaş

Keyword(s):

Integer Sequences ◽

Lucas Sequences ◽

Lucas Sequence ◽

Necessary And Sufficient

Abstract In this paper we introduce the functions which count the number of generalized Lucas and Pell-Lucas sequence terms not exceeding a given value x and, under certain conditions, we derive exact formulae (Theorems 3 and 4) and establish asymptotic limits for them (Theorem 6). We formulate necessary and sufficient arithmetic conditions which can identify the terms of a-Fibonacci and a-Lucas sequences. Finally, using a deep theorem of Siegel, we show that the aforementioned sequences contain only finitely many perfect powers. During the process we also discover some novel integer sequences.

Download Full-text