**Volume 24** ▶ Number 1 ▷ Number 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

*X*

^{4}+

*Y*

^{4}= 2

*U*

^{4}+ Σ

^{n}_{i=1}

*T*

_{i}U^{4}

_{i}, where

*n*≥ 1, and

*T*, 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

_{i}*n*,

*T*, 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.

_{i}**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

*x*

^{2}–

*dy*

^{2}= –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

*x*

^{2}–

*dy*

^{2}= –1 is solvable in integers

*x*and

*y*, where

*d*=

*d*(

*u*

_{n}, u_{n+1},

*m*) can be expressed as rational functions of

*u*and

_{n}*u*

_{n+1}and fourth-degree polynomials of m, and u

_{n}satisfies a recurrence relation:

*u*

_{0}=

*u*

_{1}= 1 and

*u*

_{n+2}= 3

*u*

_{n+1}–

*u*for

_{n}*n*∈ ℕ ∪ {0}. Our main argument is based on a binary quadratic relation between

*u*and

_{n}*u*

_{n+1}and properties 1+

*u*

*/*

_{n2}*u*

_{n+1}∈ N and 1+

*u*

^{2}

_{n+1}/

*u*∈ N. Due to the recurrence relation of

_{n}*u*, such d’s are easy to be generated by hand calculation and computational mathematics via a class of explicit formulas. Besides, we consider equation

_{n}*x*

^{2}–

*k*(

*k*+ 4)

*m*

^{2}

*y*

^{2}= –1 and show that it is solvable in integers if and only if

*k*= 1 and

*m*∈ ℕ is a divisor of

*u*

_{3n+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

*y*

^{2}=

*x*

^{3}–

*A*

^{2}

*x*+

*B*

^{2}, 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

*y*

^{2}=

*x*(

*x*–

*a*

^{2})(

*x*+

*c*

^{2}) and

*y*

^{2}=

*x*(

*x*–

*b*

^{2})(

*x*+

*c*

^{2}), 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

*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

**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

_{k=1}

^{n}(𝑎

^{2}𝑘

^{4}+ (2𝑎 − 𝑎

^{2})𝑘

^{2}+ 1) = 𝑦

^{2}. This Diophantine equation generalizes a result of Gürel [5] for 𝑎 = 2. We also prove that the product (2

^{2}− 1)(3

^{2}− 1)…(𝑛

^{2}− 1) is a perfect square only for the values 𝑛 for which the triangular number T

^{n}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

^{𝑘}𝑛

^{2}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

_{𝑓}such that 𝑓(𝑚𝑛) = 𝑓(𝑚)ℎ

_{𝑓}(𝑛) + 𝑓(𝑛)ℎ

_{𝑓}(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 ℎ

_{𝐷}(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 ℎ

_{𝑓}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

*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

_{𝑥→∞}𝐴(𝑥) = 1.08366…, which is the Legendre’s constant. It is well-known that lim

_{𝑥→∞}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

**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

^{𝑛})

^{𝑛≥0}be the Lucas sequence. D. Marques and A. Togbé [7] showed that if 𝐹

_{𝑛}…𝐹

_{𝑛+𝑘−1}is a repdigit with at least two digits, then (𝑘, 𝑛) = (1, 10), where (𝐹

_{𝑛})

_{≥0}is the

Fibonacci sequence. In this paper, we solve the equation

𝐿

_{𝑛}…𝐿

_{𝑛+𝑘−1}= 𝑎 (︂10

^{𝑚}− 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

**On two new combined 3-Fibonacci sequences. Part 2**

*Original research paper. Pages 111–114*

Krassimir T. Atanassov

Full paper (PDF, 134 Kb) | Abstract

**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

_{𝑛}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

^{−}(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

*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*:

*V*→

*R*the weight of is The Roman Domination Number (RDN) denoted by

*γ*(

_{R }*G*) is the minimum weight among all RDF in

*G*. If

*V*–

*D*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

*γ*

^{1}

*(*

_{R}*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.*