Volume 24, 2018, Number 3

Volume 24Number 1Number 2 ▷ Number 3 ▷ Number 4

On a class of quartic Diophantine equations of at least five variables
Original research paper. Pages 1–9
Hamid Reza Abdolmalki and Farzali Izadi
Full paper (PDF, 179 Kb) | Abstract

In this paper, elliptic curves theory is used for solving the quartic Diophantine equation X4 + Y4 = 2U4 + Σni=1TiU4i, where n ≥ 1, and Ti, are rational numbers. We try to transform this quartic to a cubic elliptic curve of positive rank, then get infinitely many integer solutions for the aforementioned Diophantine equation. We solve the above Diophantine equation for some values of n, Ti, and obtain infinitely many nontrivial integer solutions for each case. We show among the other things that some numbers can be written as sums of some biquadrates in two different ways with different coefficients.

On negative Pell equations: Solvability and unsolvability in integers
Original research paper. Pages 10–26
Hsin-Te Chiang, Mei-Ru Ciou, Chia-Ling Tsai, Yuh-JennWu and Chiun-Chang Lee
Full paper (PDF, 248 Kb) | Abstract

Solvability criteria of negative Pell equations x2dy2 = –1 have previously been established via calculating the length for the period of the simple continued fraction of √d and checking the existence of a primitive Pythagorean triple for d. However, when d » 1, such criteria usually require a lengthy calculation. In this note, we establish a novel approach to construct integers d such that x2dy2 = –1 is solvable in integers x and y, where d = d(un, un+1,m) can be expressed as rational functions of un and un+1 and fourth-degree polynomials of m, and un satisfies a recurrence relation: u0 = u1 = 1 and un+2 = 3un+1un for n ∈ ℕ ∪ {0}. Our main argument is based on a binary quadratic relation between un and un+1 and properties 1+un2 / un+1 ∈ N and 1+u2n+1 / un ∈ N. Due to the recurrence relation of un, such d’s are easy to be generated by hand calculation and computational mathematics via a class of explicit formulas. Besides, we consider equation x2k(k + 4)m2y2 = –1 and show that it is solvable in integers if and only if k = 1 and m ∈ ℕ is a divisor of u3n+2 for some n ∈ ℕ ∪ {0}. The main approach for its solvability is the Fermat’s method of infinite descent.

Some new families of positive-rank elliptic curves arising from Pythagorean triples
Original research paper. Pages 27–36
Mehdi Baghalaghdam and Farzali Izadi
Full paper (PDF, 193 Kb) | Abstract

In the present paper, we introduce some new families of elliptic curves with positive rank arising from Pythagorean triples. We study elliptic curves of the form y2 = x3A2x + B2, where A, B ∈ {a, b, c} are two different numbers and (a, b, c) is a rational Pythagorean triple. First of all, we prove that if (a, b, c) is a primitive Pythagorean triple (PPT), then the rank of each family is positive. Furthermore, we construct subfamilies of rank at least 3 in each family but one with rank at least 2, and obtain elliptic curves of high rank in each family. Finally, we consider two other new families of elliptic curves of the forms y2 = x(xa2)(x + c2) and y2 = x(xb2)(x + c2), and prove that if (a, b, c) is a PPT, then the rank of each family is positive.

On the derivatives of bivariate Fibonacci polynomials
Original research paper. Pages 37–46
Tuba Çakmak and Erdal Karaduman
Full paper (PDF, 169 Kb) | Abstract

In this study, the new algebraic properties related to bivariate Fibonacci polynomials have been given. We present the partial derivatives of these polynomials in the form of convolution of bivariate Fibonacci polynomials. Also, we define a new recurrence relation for the r-th partial derivative sequence of bivariate Fibonacci polynomials.

2-Fibonacci polynomials in the family of Fibonacci numbers
Original research paper. Pages 47–55
Engin Özkan, Merve Taştan and Ali Aydoğdu
Full paper (PDF, 206 Kb) | Abstract

In the present study, we define new 2-Fibonacci polynomials by using terms of a new family of Fibonacci numbers given in [4]. We show that there is a relationship between the coefficient of the 2-Fibonacci polynomials and Pascal’s triangle. We give some identities of the 2-Fibonacci polynomials. Afterwards, we compare the polynomials with known Fibonacci polynomials. We also express 2-Fibonacci polynomials by the Fibonacci polynomials. Furthermore, we prove some theorems related to the polynomials. Also, we introduce the derivative of the 2-Fibonacci polynomials.

On products of quartic polynomials over consecutive indices which are perfect squares
Original research paper. Pages 56–61
Kantaphon Kuhapatanakul, Natnicha Meeboomak and Kanyarat Thongsing
Full paper (PDF, 146 Kb) | Abstract

Let 𝑎 be a positive integer. We study the Diophantine equation Π︁k=1n(𝑎2𝑘4 + (2𝑎 − 𝑎2)𝑘2 + 1) = 𝑦2. This Diophantine equation generalizes a result of Gürel [5] for 𝑎 = 2. We also prove that the product (22 − 1)(32 − 1)…(𝑛2 − 1) is a perfect square only for the values 𝑛 for which the triangular number Tn is a perfect square.

Conditions equivalent to the Descartes–Frenicle–Sorli Conjecture on odd perfect numbers – Part II
Original research paper. Pages 62–67
Jose Arnaldo Bebita Dris and Doli-Jane Uvales Tejada
Full paper (PDF, 156 Kb) | Abstract

The Descartes–Frenicle–Sorli conjecture predicts that 𝑘 = 1 if 𝑞𝑘𝑛2 is an odd perfect number with Euler prime 𝑞. In this note, we present some further conditions equivalent to this conjecture.

The arithmetic derivative and Leibniz-additive functions
Original research paper. Pages 68–76
Pentti Haukkanen, Jorma K. Merikoski and Timo Tossavainen
Full paper (PDF, 162 Kb) | Abstract

An arithmetic function 𝑓 is Leibniz-additive if there is a completely multiplicative function ℎ𝑓 such that 𝑓(𝑚𝑛) = 𝑓(𝑚)ℎ𝑓(𝑛) + 𝑓(𝑛)ℎ𝑓(𝑚) for all positive integers 𝑚 and 𝑛. A motivation for the present study is the fact that Leibniz-additive functions are generalizations of the arithmetic derivative 𝐷; namely, 𝐷 is Leibniz-additive with ℎ𝐷(𝑛) = 𝑛. We study the basic properties of Leibniz-additive functions and, among other things, show that a Leibniz-additive function 𝑓 is totally determined by the values of 𝑓 and ℎ𝑓 at primes. We also find connections of Leibniz-additive functions to the usual product, composition and Dirichlet convolution of arithmetic functions. The arithmetic partial derivative is also considered.

Prime sequences
Original research paper. Pages 77–83
J. V. Leyendekkers and A. G. Shannon
Full paper (PDF, 87 Kb) | Abstract

Primes are considered in three sequences, of which two are exclusive to specific primes. These sequences have the integers represented in the form nR where R is the right-end-digit of the prime and n represents the remaining left digits which are given by linear equations.

One more disproof for the Legendre’s conjecture regarding the prime counting function 𝜋(𝑥)
Original research paper. Pages 84–91
Reza Farhadian and Rafael Jakimczuk
Full paper (PDF, 187 Kb) | Abstract

Let 𝜋(𝑥) denote the prime counting function, i.e., the number of primes not exceeding 𝑥. The Legendre’s conjecture regarding the prime counting function states that 𝜋(𝑥) = 𝑥 / (log 𝑥 − 𝐴(𝑥)), where Legendre conjectured that lim𝑥→∞𝐴(𝑥) = 1.08366…, which is the Legendre’s constant. It is well-known that lim𝑥→∞𝐴(𝑥) = 1, and hence the Legendre’s conjecture is not true. In this article we give various proofs of this limit and establish some generalizations.

An inequality involving a ratio of zeta functions
Original research paper. Pages 92–94
József Sándor
Full paper (PDF, 136 Kb) | Abstract

We prove an inequality for a ratio of zeta functions. This extends a classical result (see [2]). The method is based on Dirichlet series, combined with real analysis.

On repdigits as product of consecutive Lucas numbers
Original research paper. Pages 95–102
Nurettin Irmak and Alain Togbé
Full paper (PDF, 158 Kb) | Abstract

Let (𝐿𝑛)𝑛≥0 be the Lucas sequence. D. Marques and A. Togbé [7] showed that if 𝐹𝑛…𝐹𝑛+𝑘−1 is a repdigit with at least two digits, then (𝑘, 𝑛) = (1, 10), where (𝐹𝑛)≥0 is the
Fibonacci sequence. In this paper, we solve the equation
𝐿𝑛…𝐿𝑛+𝑘−1 = 𝑎 (︂10𝑚 − 1) / 9, where 1 ≤ 𝑎 ≤ 9, 𝑛, 𝑘 ≥ 2 and 𝑚 are positive integers.

Generalized golden ratios and associated Pell sequences
Original research paper. Pages 103–110
A. G. Shannon and J. V. Leyendekkers
Full paper (PDF, 105 Kb) | Abstract

This paper considers generalizations of the golden ratio based on an extension of the Pell recurrence relation. These include related partial difference equations. It develops generalized Pell and Companion-Pell numbers and shows how they can yield elegant generalizations of Fibonacci and Lucas identities. This sheds light on the format of the original identities, such as the Simson formula, to distinguish what is significant and substantial from what is incidental or accidental.

On two new combined 3-Fibonacci sequences. Part 2
Original research paper. Pages 111–114
Krassimir T. Atanassov
Full paper (PDF, 134 Kb) | Abstract

Two new combined 3-Fibonacci sequences are introduced and the explicit formulae for their 𝑛-th members are given.

Enumeration of 3- and 4-Wilf classes of four 4-letter patterns
Original research paper. Pages 115–130
David Callan and Toufik Mansour
Full paper (PDF, 221 Kb) | Abstract

Let 𝑆𝑛 be the symmetric group of all permutations of 𝑛 letters. We show that there are precisely 27 (respectively, 15) Wilf classes consisting of exactly 3 (respectively, 4) symmetry classes of subsets of four 4-letter patterns.

Embedding of signed regular graphs
Original research paper. Pages 131–141
Deepa Sinha and Anita Kumari Rao
Full paper (PDF, 211 Kb) | Abstract

A signed graph is a graph whose edges carry the weight ‘+’ or ‘−’. A signed graph 𝑆 is called signed-regular if 𝑑(𝑣) is same for all 𝑣 ∈ 𝑉 and 𝑑+(𝑣) is same for all 𝑣 ∈ 𝑉. The problems of embedding (𝑖, 𝑗)-signed-regular graphs in (𝑖, 𝑗 + 𝑙)-signed-regular graphs is one of the fascinating problems from application point of view, which is dealt in this paper with insertion of least number of vertices in 𝑆.

Roman and inverse Roman domination in graphs
Original research paper. Pages 142–150
Zulfiqar Zaman, M. Kamal Kumar and Saad Salman Ahmad
Full paper (PDF, 229 Kb) | Abstract

Motivated by the article in Scientific American [8], Michael A. Henning and Stephen T. Hedetniemi explored the strategy of defending the Roman Empire. Cockayne defined Roman dominating function (RDF) on a Graph G = (V, E) to be a function f : V → {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0.  is adjacent to at least one vertex v for which f (v) = 2. For a real valued function f : VR  the weight of  is  The Roman Domination Number (RDN) denoted by γR (G) is the minimum weight among all RDF in G. If VD contains a Roman dominating function  f 1 : V → {0, 1, 2}, where D is the set of vertices v for which f (v) > 0. Then f 1 is called inverse Roman dominating function (IRDF) on a graph G w.r.t. f. The inverse Roman domination number (IRDN) denoted by γ1R(G) is the minimum weight among all IRDF in G. In this paper we find few results of RDN and IRDN.

This issue of the International Journal “Notes on Number Theory and Discrete Mathematics” is published with the financial support of the Bulgarian National Science Fund, Grant Ref. No. DNP-06-38/2017.

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