**Volume 27** ▶ Number 1

**On a class of quartic Diophantine equations**

*Original research paper. Pages 1—6*

F. Izadi, M. Baghalaghdam and S. Kosari

Full paper (PDF, 177 Kb) | Abstract

**On the k-Mersenne–Lucas numbers**

*Original research paper. Pages 7—13*

Mourad Chelgham and Ali Boussayoud

Full paper (PDF, 202 Kb) | Abstract

*k*-Mersenne–Lucas numbers and investigate some properties. Then, we obtain some identities and established connection formulas between

*k*-Mersenne–Lucas numbers and

*k*-Mersenne numbers through the use of Binet’s formula.

**On Münchhausen numbers**

*Original research paper. Pages 14—21*

Miroslav Kureš

Full paper (PDF, 266 Kb) | Abstract

^{3}+ 4

^{4}+ 3

^{3}+ 5

^{5}, was formalized to the notion of Münchhausen numbers. Properties of these numbers are studied and it is showed that although there are only finitely many Münchhausen numbers in a given base

*b*, there are infinitely many Münchhausen numbers of the length 2 in all bases. A certain reversion in the definition gives the notion of so-called anti-Munchhausen numbers, search for them is computationally more effective.

**On new arithmetic function relative to a fixed positive integer. Part 1**

*Original research paper. Pages 22—26*

Brahim Mittou and Abdallah Derbal

Full paper (PDF, 154 Kb) | Abstract

**Odd/even cube-full numbers**

*Original research paper. Pages 27—31*

Tippawan Puttasontiphot and Teerapat Srichan

Full paper (PDF, 148 Kb) | Abstract

^{−1/3}+ 2

^{−2/3}.

**A study on some generalized multiplicative and generalized additive arithmetic functions**

*Original research paper. Pages 32—44*

D. Bhattacharjee

Full paper (PDF, 600 Kb) | Abstract

*-function), an additive function and a generalized additive function (or briefly a*

**GM***-function). We shall study in details some properties of*

**GA***-functions as well as*

**GM***-functions using some particular arithmetic convolutions namely the Narkiewicz’s*

**GA***A*-product and the author’s

*B*-product. We conclude our discussion with some examples.

**Bi-unitary multiperfect numbers, IV(b)**

*Original research paper. Pages 45—69*

Pentti Haukkanen and Varanasi Sitaramaiah

Full paper (PDF, 297 Kb) | Abstract

Peter Hagis (1987) proved that there are no odd bi-unitary multiperfect numbers. The present paper is part IV(b) in a series of papers on even bi-unitary multiperfect numbers. In parts I, II and III we considered bi-unitary triperfect numbers of the form , where and is odd. In part IV(a) we solved partly the case . We proved that if is a bi-unitary triperfect number of the form , where , then . We then solved completely the case . In the present paper we give some partial results concerning the case under the assumption .

**On a Diophantine equation arising in the history of mathematics**

*Original research paper. Pages 70—75*

József Sándor and Krassimir Atanassov

Full paper (PDF, 136 Kb) | Abstract

**Finiteness of lattice points on varieties F(y) = F(g(𝕏)) + r(𝕏) over imaginary quadratic fields**

*Original research paper. Pages 76—90*

Lukasz Nizio

Full paper (PDF, 283 Kb) | Abstract

**A note on Q-matrices and higher order Fibonacci polynomials**

*Original research paper. Pages 91—100*

Paolo Emilio Ricci

Full paper (PDF, 219 Kb) | Abstract

*Q*-matrices are extended to the case of Fibonacci, Tribonacci and R-bonacci polynomials.

**The Oresme sequence: The generalization of its matrix form and its hybridization process**

*Original research paper. Pages 101—111*

Milena Carolina dos Santos Mangueira, Renata Passos Machado Vieira, Francisco Regis Vieira Alves and Paula Maria Machado Cruz Catarino

Full paper (PDF, 230 Kb) | Abstract

**The character sum of polynomials with k variables and two-term exponential sums**

*Original research paper. Pages 112—124*

Xu Xiaoling, Zhang Jiafan and Zhang Wenpeng

Full paper (PDF, 201 Kb) | Abstract

**A very general binomial matrix**

*Original research paper. Pages 125—133*

Leo Betthauser, Ömür Deveci and Josh Hiller

Full paper (PDF, 181 Kb) | Abstract

**How to sum powers of balancing numbers efficiently**

*Original research paper. Pages 134—137*

Helmut Prodinger

Full paper (PDF, 145 Kb) | Abstract

**Characterization of ideals of rhotrices over a ring and its applications**

*Original research paper. Pages 138—147*

Kailash M. Patil

Full paper (PDF, 236 Kb) | Abstract

**On the connections between Pell numbers and Fibonacci p-numbers**

*Original research paper. Pages 148—160*

Anthony G. Shannon, Özgür Erdağ and Ömür Deveci

Full paper (PDF, 235 Kb) | Abstract

*p*-sequence and then we discuss the connection of the Fibonacci–Pell

*p*-sequence with the Pell and Fibonacci

*p*-sequences. Also, we provide a new Binet formula and a new combinatorial representation of the Fibonacci–Pell

*p*-numbers by the aid of the

*n*-th power of the generating matrix of the Fibonacci–Pell

*p*-sequence. Furthermore, we derive relationships between the Fibonacci–Pell

*p*-numbers and their permanent, determinant and sums of certain matrices.

**A note on Mersenne Padovan and Perrin numbers**

*Original research paper. Pages 161—170*

Bir Kafle, Salah Eddine Rihane and Alain Togbé

Full paper (PDF, 215 Kb) | Abstract

**Pell–Padovan generalized quaternions**

*Original research paper. Pages 171—187*

Zehra İşbilir and Nurten Gürses

Full paper (PDF, 247 Kb) | Abstract

**Triple sums including Fibonacci numbers with three binomial coefficients**

*Original research paper. Pages 188—197*

Funda Taşdemir

Full paper (PDF, 179 Kb) | Abstract

**Catalan transform of the k-Pell, k-Pell–Lucas and modified k-Pell sequence**

*Original research paper. Pages 198—207*

Merve Taştan and Engin Özkan

Full paper (PDF, 865 Kb) | Abstract

*k*-Pell sequence, the

*k*-Pell–Lucas sequence and the Modified

*k*-Pell sequence and examine the properties of the sequences. Then we apply the Hankel transform to the Catalan transforms of the

*k*-Pell sequence, the Catalan transform of the

*k*-Pell–Lucas sequence and the Catalan transform of the Modified

*k*-Pell sequence. Also, we obtain the generating functions of the Catalan transform of the

*k*-Pell sequence,

*k*-Pell–Lucas sequence and Modified

*k*-Pell sequence. Furthermore, we acquire an interesting characteristic related to the determinant of the Hankel transform of the sequences.

**On the spectra of a new duplication based corona of graphs**

*Original research paper. Pages 208—220*

Renny P. Varghese and D. Susha

Full paper (PDF, 201 Kb) | Abstract

*duplication corresponding corona*. Here we mainly determine the adjacency, Laplacian and signless Laplacian spectra of the new graph product. In addition to that, we find out the incidence energy, the number of spanning trees, Kirchhoff index and Laplacian-energy-like invariant of the new graph. Also we discuss some new classes of cospectral graphs.

*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. KP-06-NP2/26/2020.*

**Volume 27** ▶ Number 1