**Volume 25** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**The linear combination of two triangular numbers is a perfect square**

*Original research paper. Pages 1–12*

Junyao Peng

Full paper (PDF, 187 Kb) | Abstract

*n*is not a perfect square, the Diophantine equation

has infinitely many positive integer solutions (*y,z*). Second, we prove that if *m,n* are some special values, the Diophantine equation

**Composition of happy functions**

*Original research paper. Pages 13–20*

Passawan Noppakeaw, Niphawan Phoopha and Prapanpong Pongsriiam

Full paper (PDF, 193 Kb) | Abstract

if is the expansion of in base . We call an -happy function. Let be a composition of various -happy functions. We show that, for any given , the iteration sequence either converges to a fixed point or eventually becomes a cycle. Here is the identity function mapping to for all and is the -fold composition of . In addition, we prove that the number of all possible fixed points and cycles is finite. Examples are also given.

**Direct parametrization of Pythagorean triples**

*Original research paper. Pages 21–35*

Sungkon Chang

Full paper (PDF, 240 Kb) | Abstract

**Extension factor: definition, properties and problems. Part 1**

*Original research paper. Pages 36–43*

Krassimir T. Atanassov and József Sándor

Full paper (PDF, 166 Kb) | Abstract

**Numbers with the same kernel**

*Original research paper. Pages 44–64*

Rafael Jakimczuk

Full paper (PDF, 241 Kb) | Abstract

where

and the positive coefficients of the series have a strong connection with the prime numbers. We also prove that

where . The methods used are very elementary. The case , namely , was studied, as it is well-known, by N. G. de Bruijn (1962) and W. Schwarz (1965).

**On ternary biquadratic Diophantine equation 11( x^{2} – y^{2}) + 3(x + y ) =10z^{4} **

*Original research paper. Pages 65–71*

S. Vidhyalakshmi, M. A. Gopalan , S. A. Thangam and Ö. Özer

Full paper (PDF, 91 Kb) | Abstract

*x, y, z*) satisfying the non-homogeneous bi-quadratic equation with three unknowns 11(

*x*

^{2}–

*y*

^{2}) + 3(

*x*+

*y*) =10

*z*

^{4}. Various interesting properties among the values of

*x, y, z*are presented. Some relations between the solutions and special numbers are exhibited.

**On applications of blending generating functions of q-Apostol-type polynomials**

*Original research paper. Pages 72–86*

Ugur Duran, Mehmet Acikgoz and Serkan Araci

Full paper (PDF, 217 Kb) | Abstract

*q*-Stirling numbers of the second kind. Additionally, by making use of the fermionic

*p*-adic integral over the

*p*-adic numbers field, some relationships including unified Apostol-type

*q*-polynomials and classical Euler numbers are obtained.

**Gaussian Mersenne numbers and generalized Mersenne quaternions**

*Original research paper. Pages 87–96*

Ahmet Daşdemir and Göksal Bilgici

Full paper (PDF, 156 Kb) | Abstract

**A note on generalized Leonardo numbers**

*Original research paper. Pages 97–101*

A. G. Shannon

Full paper (PDF, 98 Kb) | Abstract

**Sum formulas involving powers of balancing and Lucas-balancing numbers – II**

*Original research paper. Pages 102–110*

S. G. Rayaguru and G. K. Panda

Full paper (PDF, 146 Kb) | Abstract

**Dual-complex k-Pell quaternions**

*Original research paper. Pages 111–125*

Fügen Torunbalcı Aydın

Full paper (PDF, 200 Kb) | Abstract

*k*-Pell numbers and dual-complex

*k*-Pell quaternions are defined. Also, some algebraic properties of dual-complex

*k*-Pell numbers and quaternions which are connected with dual-complex numbers and

*k*-Pell numbers are investigated. Furthermore, Honsberger Identity, d’Ocagne’s Identity, Binet’s Formula, Cassini’s Identity and Catalan’s Identity for these quaternions are given.

**On bicomplex numbers with coefficients from the complex Fibonacci sequence**

*Original research paper. Pages 126–137*

Serpil Halıcı and Şule Çürük

Full paper (PDF, 182 Kb) | Abstract

tal properties of the newly defined sequence.

**A remark on the Tribonacci sequences**

*Original research paper. Pages 138–141*

Lilija Atanassova

Full paper (PDF, 110 Kb) | Abstract

**Convolution identities for Tetranacci numbers**

*Original research paper. Pages 142–169*

Takao Komatsu and Rusen Li

Full paper (PDF, 245 Kb) | Abstract

**Close encounters of the golden and silver ratios**

*Original research paper. Pages 170–184*

Robin James Spivey

Full paper (PDF, 233 Kb) | Abstract

**Sequences of Tridovan and their identities**

*Original research paper. Pages 185–197*

Renata Passos Machado Vieira, and Francisco Regis Vieira Alves

Full paper (PDF, 197 Kb) | Abstract

**On the software computation of the formulae for the n-th prime number**

*Original research paper. Pages 198–206*

Dimitar G. Dimitrov

Full paper (PDF, 188 Kb) | Abstract

*n*-th prime number exist. In this paper, a comparison of the computation time of different existing formulae is made.

**On the distribution of k-free numbers and r-tuples of k-free numbers. A survey**

*Survey paper. Pages 207–222*

Radoslav Tsvetkov

Full paper (PDF, 226 Kb) | Abstract

*k*-free numbers and

*r*-tuples of

*k*-free numbers. We state the main problems in the field, sketch their history and the basic machinery used to study them.

*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-NP-28/2018.*