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

**Sums of reciprocals of integers**

*Original research paper. Pages 1–7*

Simon Davis

Full paper (PDF, 182 Kb) | Abstract

**Improving the estimates for a sequence involving prime numbers**

*Original research paper. Pages 8–13*

Christian Axler

Full paper (PDF, 146 Kb) | Abstract

*C*,

_{n}= np_{n}− ∑_{k ≤ n}p_{k}*n*≥ 1, involving the prime numbers.

**New methods for obtaining new families of congruent numbers**

*Original research paper. Pages 14–24*

Hamid Reza Abdolmalki and Farzali Izadi

Full paper (PDF, 191 Kb) | Abstract

*p*= 8

*k*+ 1 are CNs. Among other things, we also introduce two simple methods to find some CNs of the forms p ≡ 1 (mod 8) and 2

*p*where

*p*is a prime number. By non-CNs and our methods, we also obtain some Diophantine equations (especially of degree 4), which have no positive solutions. In the end, we obtain a result on Heron triangles.

**Larger families of solutions to some Diophantine equations**

*Original research paper. Pages 25–31*

Lyes Ait-Amrane

Full paper (PDF, 138 Kb) | Abstract

**A note on Euler’s totient function**

*Original research paper. Pages 32–35*

József Sándor

Full paper (PDF, 157 Kb) | Abstract

^{k}+ 1) > 2

^{k −1}and ϕ(2

^{m}+ 1) < 2

^{m − 1}both have infinitely many solutions.

**Minimal sets of shifted values of the Euler totient function**

*Original research paper. Pages 36–47*

Martin Kreh and Katrin Neuenstein

Full paper (PDF, 193 Kb) | Abstract

*a*for 1 ≤

*a*≤ 5.

**New form of the Newton’s binomial theorem**

*Original research paper. Pages 48–49*

Mladen V. Vassilev-Missana

Full paper (PDF, 119 Kb) | Abstract

**Inequalities between the arithmetic functions φ, ψ and σ. Part 1**

*Original research paper. Pages 50–53*

Krassimir T. Atanassov and József Sándor

Full paper (PDF, 149 Kb) | Abstract

*φ*,

*ψ*and

*σ*are proved the inequalities

*ψ*(

*n*)

^{n}>

*σ*(

*n*)

^{φ(n)}and

*σ*(

*n*)

^{n}<

*ψ*(

*n*)

^{σ(n)}for each natural number

*n*≥ 2.

**Partitions generated by Mock Theta Functions ρ(q), σ(q) and ν(q) and relations with partitions into distinct parts**

*Original research paper. Pages 54–74*

Alessandro Bagatini, Marília Luiza Matte and Adriana Wagner

Full paper (PDF, 252 Kb) | Abstract

*ρ*(

*q*),

*σ*(

*q*) and

*ν*(

*q*) introduced in [5], we have obtained identities for the partitions generated by their respective general terms, whose proofs are done in a completely combinatorial way. We have also obtained relations between partitions into two colours generated by

*ρ*(

*q*) and

*σ*(

*q*), and also by

*ν*(

*q*).

**Complete solving the quadratic equation mod 2 ^{n}**

*Original research paper. Pages 75–83*

S. M. Dehnavi, M. R. Mirzaee Shamsabad and A. Mahmoodi Rishakani

Full paper (PDF, 213 Kb) | Abstract

*ax*

^{2}+

*bx*+

*c*= 0 (mod 2

^{n}), and provide a complete analysis of it. More precisely, we determine when this equation has a solution and in the case that it has a solution, we give not only the number of solutions, but also the set of solutions, in

*O*(

*n*) time. One of the interesting results of our research is that, if this equation has a solution, then the number of solutions is a power of two. Most notably, as an application, we characterize the number of fixed-points of quadratic permutation polynomials over ℤ

_{2n}, which are used in symmetric cryptography.

**On Terai’s exponential equation with two finite integer parameters**

*Original research paper. Pages 84–107*

Takafumi Miyazaki

Full paper (PDF, 247 Kb) | Abstract

*r*be an integer with

*r*> 1, and

*m*be an even positive integer. Define integers

*A*and

*B*by the equation

*A*+

*B*√−1 = (

*m*+ √−1)

^{r}. It is proven by F. Luca in 2012 that the equation |

*A*|

^{x}+ |

*B*|

^{y}= (

*m*

^{2}+ 1)

^{z}does not hold for any triple (

*x*,

*y*,

*z*) of positive integers not equal to (2, 2,

*r*), whenever

*r*or

*m*exceeds some effectively computable absolute constant. In our previous work, we estimated this constant explicitly. Here that estimate is substantially improved.

**Almost balancing, triangular and square triangular numbers**

*Original research paper. Pages 108–121*

Ahmet Tekcan

Full paper (PDF, 196 Kb) | Abstract

**Averages of the Dirichlet convolution of the binary digital sum**

*Original research paper. Pages 122–127*

Teerapat Srichan

Full paper (PDF, 144 Kb) | Abstract

*s*

_{2}(

*n*), the sum of digits ofthe expansion of

*n*in base 2. The Trollope–Delange formula is used in our proof. It provides an explicit asymptotic formula for the total number of digits ‘1’ in the binary expansions of the integers between 1 and

*n*− 1 in term of the continuous, nowhere differentiable Takagi function. Moreover, we also extend the result to averages of the

*k*-th convolution of the binary digital sum, for

*k*≥ 2.

**Bijective proofs involving chromatic overpartitions**

*Original research paper. Pages 128–136*

Mateus Alegri

Full paper (PDF, 171 Kb) | Abstract

**On dual Horadam octonions**

*Original research paper. Pages 137–149*

Nayil Kılıç

Full paper (PDF, 294 Kb) | Abstract

**Power GCD and power LCM matrices defined on GCD-closed sets over unique factorization domains**

*Original research paper. Pages 150–166*

Y. A. Awad, T. Kadri and R. H. Mghames

Full paper (PDF, 225 Kb) | Abstract

*T*= {

*t*

_{1},

*t*

_{2}, …,

*t*} be a well ordered set of

_{m}*m*distinct positive integers with

*t*

_{1}<

*t*

_{2}< … <

*t*. The GCD matrix on

_{m}*T*is defined as (

*T*)

_{m×m}= (

*t*), where (

_{i}, t_{j}*t*) is the greatest common divisor of

_{i}, t_{j}*t*and

_{i}*t*, and the power GCD matrix on

_{j}*T*is (

*T*)

^{r}_{m×m}= (

*t*)

_{i}, t_{j}^{r}, where

*r*is any real number. The LCM matrix on

*T*is defined as [

*T*]

_{m×m}= [

*t*], where [

_{i}, t_{j}*t*] is the least common multiple of

_{i}, t_{j}*t*and

_{i}*t*, and the power LCM matrix on

_{j}*T*is [

*T*]

^{r}_{m×m}= [

*t*]

_{i}, t_{j}^{r}. Set

*T*= {

*t*

_{1},

*t*

_{2}, …,

*t*} is said to be gcd-closed if (

_{m}*t*) ∈

_{i}, t_{j}*T*for every

*t*in

_{i}and t_{j}*T*. In this paper, we give a generalization for the power GCD and LCM matrices defined on gcd-closed sets over unique factorization domains (UFDs). Moreover, we present a speculation for a generalization of Bourque–Ligh conjecture to UFDs which states that the least common multiple matrix defined on a gcd-closed

*P*-ordered set in any UFD is nonsingular. Some examples that show what is done are additionally given in ℤ[

*i*] and ℤ

_{p}[

*x*].

**The identities for generalized Fibonacci numbers via orthogonal projection**

*Original research paper. Pages 167–177*

Yasemin Alp and E. Gökçen Koçer

Full paper (PDF, 171 Kb) | Abstract

*R*(

*p*, 1) of generalized Fibonacci sequences and orthogonal bases of this space. Using these orthogonal bases, we obtain the orthogonal projection onto a subspace

*R*(

*p*, 1) of ℝ

^{n}. By using the orthogonal projection, we obtain the identities for the generalized Fibonacci numbers.

**Relations for generalized Fibonacci and Tribonacci sequences**

*Original research paper. Pages 178–192*

Robert Frontczak

Full paper (PDF, 184 Kb) | Abstract

**Generalized Fibonacci numbers and Bernoulli polynomials**

*Original research paper. Pages 193–198*

Anthony G. Shannon, Ömür Deveci and Özgűr Erdağ

Full paper (PDF, 82 Kb) | Abstract

**A note on the OEIS sequence A228059**

*Original research paper. Pages 199–205*

Jose Arnaldo Bebita Dris and Doli-Jane Uvales Tejada

Full paper (PDF, 153 Kb) | Abstract

*p*

^{1+4k}

*r*

^{2}, where

*p*is prime of the form 1+4

*m*,

*r*> 1, and gcd(

*p, r*) = 1 that are closer to being perfect than previous terms. In this note, we present the prime factorizations of the first 37 terms.

**Representation of higher even-dimensional rhotrix**

*Original research paper. Pages 206–219*

A. O. Isere

Full paper (PDF, 187 Kb) | Abstract

**Corrigendum to “Study of some equivalence classes of primes” [Notes on Number Theory and Discrete Mathematics, Vol. 23, 2017, No. 2, 21–29]**

*Corrigendum. Page 220*

Sadani Idir

Corrigendum (PDF, 78 Kb)

*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.*