**Volume 27** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**A parametric family of quartic Thue inequalities**

*Original research paper. Pages 1–14*

Salah Eddine Rihane, Mohand Ouamar Hernane and Alain Togbé

Full paper (PDF, 249 Kb) | Abstract

**On Robin’s criterion for the Riemann Hypothesis**

*Original research paper. Pages 15–24*

Safia Aoudjit, Djamel Berkane and Pierre Dusart

Full paper (PDF, 268 Kb) | Abstract

where is the sum of the divisors of , represents the Euler–Mascheroni constant, and denotes the -fold iterated logarithm. In this note we get the following better effective estimates:

The idea employed will lead us to a possible new reformulation of the Riemann Hypothesis in terms of arithmetic functions.

**New consequences of prime-counting function**

*Original research paper. Pages 25–31*

Sadani Idir

Full paper (PDF, 204 Kb) | Abstract

**Leonardo’s three-dimensional relations and some identities**

*Original research paper. Pages 32–42*

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

Full paper (PDF, 181 Kb) | Abstract

*i*and

*j*are inserted. Finally, some three-dimensional identities are presented for Leonardo’s numbers.

**The congruence speed formula**

*Original research paper. Pages 43–61*

Marco Ripà

Full paper (PDF, 227 Kb) | Abstract

*, which is the constancy of its congruence speed for any sufficiently large*

^{b}a*b*=

*b*(

*a*). Assuming radix-10 (the well known decimal numeral system), we provide an explicit formula for the congruence speed

*V*(

*a*) ∈ ℕ

_{0}of any

*a*∈ ℕ − {0} that is not a multiple of 10. In particular, for any given

*n*∈ ℕ, we prove to be true Ripà’s conjecture on the smallest

*a*such that

*V*(

*a*) =

*n*. Moreover, for any

*a*≠ 1 ∶

*a*≢ 0 (mod 10), we show the existence of infinitely many prime numbers,

*p*=

_{j}*p*(

_{j}*V*(

*a*)), such that

*V*(

*p*) =

_{j}*V*(

*a*).

**An elementary unified approach to prove some identities involving Fibonacci and Lucas numbers**

*Original research paper. Pages 62–79*

Moussa Benoumhani

Full paper (PDF, 236 Kb) | Abstract

**Explicit formulas for Euler polynomials and Bernoulli numbers**

*Original research paper. Pages 80–89*

Laala Khaldi, Farid Bencherif and Miloud Mihoubi

Full paper (PDF, 207 Kb) | Abstract

**A simple proof of Linas’s theorem on Riemann zeta function**

*Original research paper. Pages 90–94*

Jun Ikeda, Junsei Kochiya and Takato Ui

Full paper (PDF, 142 Kb) | Abstract

**Fibonacci-Zeta infinite series associated with the polygamma functions**

*Original research paper. Pages 95–103*

Kunle Adegoke and Sourangshu Ghosh

Full paper (PDF, 172 Kb) | Abstract

**Series expansion of the Gamma function and its reciprocal**

*Original research paper. Pages 104–115*

Ioana Petkova

Full paper (PDF, 284 Kb) | Abstract

Theorem 3.1 and Theorem 3.2 are our main results. With the help of the first theorem we give our approach for finding the coefficients of Maclaurin series for and its reciprocal in an explicit form.

**Determinantal representations for the number of subsequences without isolated odd terms**

*Original research paper. Pages 116–121*

Milica Anđelic and Carlos M. da Fonseca

Full paper (PDF, 157 Kb) | Abstract

**Binomial formulas via divisors of numbers**

*Original research paper. Pages 122–128*

Karol Gryszka

Full paper (PDF, 193 Kb) | Abstract

*ω*(

*n*) counting distinct prime factors of

*n*.

**Formulas for the n-th prime number**

*Original research paper. Pages 129–139*

Krassimir T. Atanassov

Full paper (PDF, 205 Kb) | Abstract

*n*-th prime number is given and some new formulas are introduced.

**Algorithms for computing sums of powers of arithmetic progressions by using Eulerian numbers**

*Original research paper. Pages 140–148*

Peter J. Shiue, Shen C. Huang and Jorge E. Reyes

Full paper (PDF, 286 Kb) | Abstract

*et al.*‘s result with general Eulerian numbers [12]. Moreover, an analysis of theoretical time complexities is presented to show our algorithm is less complex. Various values of are analyzed in the proposed algorithms to add significance to the results. The experimental results show the proposed algorithm remains around faster as increases.

**On certain inequalities for the prime counting function**

*Original research paper. Pages 149–153*

József Sándor

Full paper (PDF, 148 Kb) | Abstract

*π*(

*x*). Particularly, a new proof and a refinement of an inequality from [1] is provided.

**A plane trigonometric proof for the case n = 4 of Fermat’s Last Theorem**

*Original research paper. Pages 154–163*

Giri Prabhakar

Full paper (PDF, 221 Kb) | Abstract

*n*= 4 of Fermat’s Last Theorem. We first show that every triplet of positive real numbers (

*a, b, c*) satisfying

*a*

^{4}+

*b*

^{4}=

*c*

^{4}forms the sides of an acute triangle. The subsequent proof is founded upon the observation that the Pythagorean description of every such triangle expressed through the law of cosines must exactly equal the description of the triangle from the Fermat equation. On the basis of a geometric construction motivated by this observation, we derive a class of polynomials, the roots of which are the sides of these triangles. We show that the polynomials for a given triangle cannot all have rational roots. To the best of our knowledge, the approach offers new geometric and algebraic insight into the irrationality of the roots.

**A note on the Neyman–Rayner triangle**

*Original research paper. Pages 164–166*

A. G. Shannon

Full paper (PDF, 72 Kb) | Abstract

**Relations on higher dimensional Padovan sequences**

*Original research paper. Pages 167–179*

Renata Passos Machado Vieira, Francisco Regis Vieira Alves and Paula Maria Machado Cruz Catarino

Full paper (PDF, 187 Kb) | Abstract

*n*-dimensional Padovan Sequence. Several mathematical properties are discussed for the first time in the present work.

**A few remarks on the values of the Bernoulli polynomials at rational arguments and some relations with ζ(2k + 1)**

*Original research paper. Pages 180–186*

André Pierro de Camargo and Giulliano Cogui de Oliveira Teruya

Full paper (PDF, 216 Kb) | Abstract

*ζ*(2

*k*+ 1) be a rational multiple of

*π*

^{2k+1}.

**On k-circulant matrices with the generalized third-order Pell numbers**

*Original research paper. Pages 187–206*

Yüksel Soykan

Full paper (PDF, 265 Kb) | Abstract

*k*-circulant matrix with the generalized third-order Pell numbers. We also study the spectral norm of this

*k*-circulant matrix. Furthermore, some numerical results for demonstrating the validity of the hypotheses of our results are given.

**Factorizations of some lower triangular matrices and related combinatorial identities**

*Original research paper. Pages 207–218*

Cahit Köme

Full paper (PDF, 264 Kb) | Abstract

*r*-eliminated Pascal matrix, Stirling matrix of the first and of the second kind matrices. We give factorizations and inverse factorizations of these matrices by virtue of the second order recurrence matrix. Moreover, we derive several combinatorial identities which are more general results of some earlier works.

**Fundamental properties of extended Horadam numbers**

*Original research paper. Pages 219–235*

Gülsüm Yeliz Şentürk, Nurten Gürses and Salim Yüce

Full paper (PDF, 302 Kb) | Abstract

**On Fibonacci quaternion matrix**

*Original research paper. Pages 236–244*

Serpil Halici and Ömür Deveci

Full paper (PDF, 205 Kb) | Abstract

**Incomplete generalized Vieta–Pell and Vieta–Pell–Lucas polynomials**

*Original research paper. Pages 245–256*

Bahar Kuloğlu, Engin Özkan and Anthony G. Shannon

Full paper (PDF, 149 Kb) | Abstract

**On k-Fibonacci hybrid numbers and their matrix representations**

*Original research paper. Pages 257–266*

Fügen Torunbalcı Aydın

Full paper (PDF, 203 Kb) | Abstract

*k*-Fibonacci hybrid numbers are defined. Also, some algebraic properties of

*k*-Fibonacci hybrid numbers such as Honsberger identity, Binet Formula, generating functions, d’Ocagne identity, Cassini and Catalan identities are investigated. In addition, we also give 2 × 2 and 4 × 4 representations of the

*k*-Fibonacci hybrid numbers

*HF*.

_{k,n}**On the dimension of an Abelian group**

*Original research paper. Pages 267–275*

Timo Tossavainen and Pentti Haukkanen

Full paper (PDF, 192 Kb) | Abstract

*G*≅ ℤ

*, dimension and rank coincide but in general they are different. For example, we show that dimension is sensitive to the overall dimensional structure of a finite or finitely generated Abelian group, whereas rank ignores the torsion subgroup completely.*

^{n}**Corrigendum to “The Oresme sequence: The generalization of its matrix form and its hybridization process” [Notes on Number Theory and Discrete Mathematics, Vol. 27, 2021, No. 1, 101–111]**

*Corrigendum. Pages 276–279*

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

Corrigendum (PDF, 140 Kb) | Abstract

*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.*