**Volume 24** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**The system of prime coordinates assigned to the positive integers**

*Original research paper. Pages 1–5*

Silviu Guiasu

Full paper (PDF, 147 Kb) | Abstract

**The ternary Goldbach problem with prime numbers of a mixed type**

*Original research paper. Pages 6–20*

S. I. Dimitrov

Full paper (PDF, 238 Kb) | Abstract

*N*can be represented in the form

*N*=

*p*

_{1}+

*p*

_{2}+

*p*

_{3},

where

*p*

_{1},

*p*

_{2},

*p*

_{3}are primes, such that

*p*

_{1}=

*x*

^{2}+

*y*

^{2}+ 1,

*p*

_{2}=

*n*.

^{c}**Alternative approach to sums of dilates**

*Original research paper. Pages 21–27*

Rafał Bystrzycki and Tomasz Schoen

Full paper (PDF, 186 Kb) | Abstract

*λ*

_{1}•

*A*+ … +

*λ*•

_{h}*A*, where

*λ*are integers. Specifically, we look for upper bounds in terms of the doubling constant

_{i}*K*= |

*A*+

*A*|/|

*A*|. We also examine some situations in which those bounds can be significantly strengthened.

**A GCD problem and a Hessenberg determinant**

*Original research paper. Pages 28–31*

M. Hariprasad

Full paper (PDF, 151 Kb) | Abstract

*a*and

*b*are coprime ((

*a*,

*b*) = 1, i.e., greatest common divisor (GCD) of

*a*and

*b*is 1), then GCD of

*a*+

*b*and (

*a*+

^{p}*b*)/(

^{p}*a*+

*b*) is either 1 or

*p*for a prime number

*p*. We prove this by linking the problem to a certain type of Hessenberg determinants.

**The double Fibonacci sequences in groups and rings**

*Original research paper. Pages 32–39*

Ömür Deveci

Full paper (PDF, 192 Kb) | Abstract

*D*

_{2m}and the ring

*E*for the generating pairs (

*a*,

*b*) and (

*b*,

*a*) as applications of the results obtained.

**On the average number of divisors of the sum of digits of squares**

*Original research paper. Pages 40–46*

Teerapat Srichan

Full paper (PDF, 196 Kb) | Abstract

*q*> 1, we present the estimation of the average number of number of divisors of the sum of digits of squares. Moreover, we extend the result to the sum of digits of the power

*h*,

*h*≥ 2.

**Methods for constructing Collatz numbers**

*Original research paper. Pages 47–54*

Abdullah N. Arslan

Full paper (PDF, 323 Kb) | Abstract

*x*; divide it by two if

*x*is even, and multiply it by 3 and add 1 if

*x*is odd; and repeat this rule on the resulting numbers, eventually we obtain 1. For a given positive integer

*x*, we say that

*x*is a Collatz number if the claim of the conjecture is true for

*x*. Computer verification reveals a large range of Collatz numbers. We develop methods by which we construct sets of Collatz numbers.

**Quotients of primes in an algebraic number ring**

*Original research paper. Pages 55–62*

Brian D. Sittinger

Full paper (PDF, 191 Kb) | Abstract

**Structural sequences for primes using right-end-digits**

*Original research paper. Pages 63–70*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 157 Kb) | Abstract

*nR*where

*R*represents the right-end-digits and

*n*represents the digits to the left of

*R*.

*n*can be classified by the sequences {3

*t*}, {3

*t*+ 1}, {3

*t*+ 2}. When

*n*= 3

*t*+ 2, no primes with

*R*= 1 or 7 can be formed with these

*n*; when

*n*= 3

*t*no primes can be formed with

*R*= 3 or 9, but when

*n*= 3

*t*+ 1, all REDs can form a prime within the constraints of imbedded sequences.

**A note on the Frobenius and the Sylvester numbers**

*Original research paper. Pages 71–73*

Amitabha Tripathi

Full paper (PDF, 138 Kb) | Abstract

**Global equitable domination in some degree splitting graphs**

*Original research paper. Pages 74–84*

S. K. Vaidya and R. M. Pandit

Full paper (PDF, 250 Kb) | Abstract

*D*of

*V*(

*G*) is called an equitable dominating set if for every

*v*∈

*V*(

*G*) −

*D*, there exists a vertex

*u*∈

*D*such that

*uv*∈

*E*(

*G*) and |

*d*(

_{G}*u*) −

*d*(

_{G}*v*)| ≤ 1. An equitable dominating set

*D*of a graph

*G*is a global equitable dominating set if it is also an equitable dominating set of the complement of

*G*. The minimum cardinality of a global equitable dominating set of

*G*is called the global equitable domination number of

*G*which is denoted by

*γ*(

^{e}_{g}*G*). We explore this concept in the context of degree splitting graphs of some graphs.

**Relationships between Fibonacci-type sequences and Golden-type ratios**

*Original research paper. Pages 85–89*

R. Patrick Vernon

Full paper (PDF, 158 Kb) | Abstract

**On two new combined 3-Fibonacci sequences**

*Original research paper. Pages 90–93*

Krassimir T. Atanassov

Full paper (PDF, 118 Kb) | Abstract

*n*-th members are given.

**Sums of powers of Fibonacci and Lucas numbers: A new bottom-up approach**

*Original research paper. Pages 94–103*

Robert Frontczak

Full paper (PDF, 165 Kb) | Abstract

**On Zudilin-like rational approximations to ζ(5)**

*Original research paper. Pages 104–116*

Anier Soria Lorente

Full paper (PDF, 209 Kb) | Abstract

*ζ*(5), after applying Zeilberger’s algorithm of creative telescoping to some hypergeometric series. These recurrence relations do not supply diophantine approximations to

*ζ*(5) that prove its irrationality, however it presents an algorithm for fast calculation of this constant. Moreover, we deduce a new continued fraction expansion for

*ζ*(5) as a consequence.

**On modular happy numbers**

*Original research paper. Pages 117–124*

Raghib Abusaris and Omar Bayyati

Full paper (PDF, 258 Kb) | Abstract

*modular happy numbers*. A number is called modular happy if the sequence obtained by iterating the process of summing the modular powers of the decimal digits of the number ends with 1.

**Even dimensional rhotrix**

*Original research paper. Pages 125–133*

A. O. Isere

Full paper (PDF, 207 Kb) | Abstract

**On the Iyengar–Madhava Rao–Nanjundiah inequality and its hyperbolic version**

*Original research paper. Pages 134–139*

József Sándor

Full paper (PDF, 155 Kb) | Abstract

this result. Certain related results are pointed out, too.

**A symmetric Diophantine equation involving biquadrates**

*Original research paper. Pages 140–144*

Ajai Choudhry

Full paper (PDF, 121 Kb) | Abstract

_{i=1}

^{n}

*a*

_{i}x_{i}^{4}= ∑

_{i=1}

^{n}

*a*

_{i}y_{i}^{4},

where

*n*≥ 3 and

*a*,

_{i}*i*= 1, 2, …,

*n*, are arbitrary nonzero integers. While a method of obtaining numerical solutions of such an equation has recently been given, it seems that an explicit parametric solution of this Diophantine equation has not yet been published. We obtain a multi-parameter solution of this equation for arbitrary values of

*a*and for any positive integer

_{i}*n*≥ 3, and deduce specific solutions when

*n*= 3 and

*n*= 4. The numerical solutions thus obtained are much smaller than the integer solutions of such equations obtained earlier.

*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.*