**Volume 21** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**“Theory of Numbers” of a complete region**

*Original research paper. Pages 1–21*

Ranjit Biswas

Full paper (PDF, 2730 Kb) | Abstract

**A variant of Waring’s problem**

*Original research paper. Pages 22–26*

Abdullah N. Arslan

Full paper (PDF, 155 Kb) | Abstract

_{i = 1}

^{n′}i

^{k}x

_{i}has at least one solution (x

_{1}, x

_{2}, …, x

_{n′}) in nonnegative integers if Σ

_{i = 1}

^{n′}i

^{k}≥ N.

**A Möbius arithmetic incidence function**

*Original research paper. Pages 27–34*

Emil Daniel Schwab and Gabriela Schwab

Full paper (PDF, 171 Kb) | Abstract

**Short remark on intuitionistic fuzziness and square-free numbers**

*Original research paper. Pages 35–37*

Krassimir T. Atanassov

Full paper (PDF, 129 Kb) | Abstract

**Improving Riemann prime counting**

*Original research paper. Pages 38–44*

Michel Planat and Patrick Solé

Full paper (PDF, 201 Kb) | Abstract

_{ρ}Ri(x

^{ρ}) depending on the critical zeros ρ of the Riemann zeta function ζ(s). We find a fit π(x) ≈ Ri

^{(3)}{ψ(x)} (with three to four new exact digits compared to li(x)) by making use of the Von Mangoldt explicit formula for the Chebyshev function ψ(x). Another equivalent fit makes use of the Gram formula with the variable ψ(x). Doing so, we evaluate π(x) in the range x = 10

^{i}, with the help of the first 2×10

^{6}Riemann zeros ρ. A few remarks related to Riemann hypothesis (RH) are given in this context.

**On some Pascal’s like triangles. Part 11**

*Original research paper. Pages 45–55*

Krassimir T. Atanassov

Full paper (PDF, 213 Kb) | Abstract

**Tridiagonal matrices related to subsequences of balancing and Lucas-balancing numbers**

*Original research paper. Pages 56–63*

Prasanta K Ray and Gopal K Panda

Full paper (PDF, 176 Kb) | Abstract

**Pell and Lucas primes**

*Original research paper. Pages 64–69*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 95 Kb) | Abstract

**Trigonometric Pseudo Fibonacci Sequence**

*Original research paper. Pages 70–76*

C. N. Phadte and S. P. Pethe

Full paper (PDF, 171 Kb) | Abstract

g

_{n+2}= g

_{n+1}+ g

_{n}+ At

^{n}, n = 0, 1, … with g

_{0}= 0, g

_{1}= 1; where both A ≠ 0 and t ≠ 0, and also t ≠ α, β where α, β are the roots of x

^{2}− x − 1 = 0.

Using the properties of generalised circular functions and Elmore’s method, we define a new sequence {H

_{n}} which is the extension of Pseudo Fibonacci Sequence, given by recurrence relation

H

_{n+2}= pH

_{n+1}− qH

_{n}+ Rt

^{n}N

_{r,0}(t

^{*}x),

where N

_{r,0}(t

^{*}x) is extended circular function.

We state and prove some properties for this extended Pseudo Fibonacci Sequence {H

_{n}} .

**Congruence properties of some partition functions**

*Original research paper. Pages 77–79*

Manvendra Tamba

Full paper (PDF, 102 Kb) | Abstract

_{3}(n)/48), where N

_{3}(n) denotes the number of ways in which n can be written as sum of three squares. We study the congruence properties of some partition

functions in relation to M(16n + 14).

**On the density of ranges of generalized divisor functions**

*Original research paper. Pages 80–87*

Colin Defant

Full paper (PDF, 173 Kb) | Abstract

_{−1}is dense in the interval . However, although the range of the function σ

_{−2}is a subset of the interval , we will see that the range of σ

_{−2}is not dense in . We begin by generalizing the divisor functions to a class of functions σ

_{t}for all real t. We then define a constant η ≈ 1.8877909 and show that if r ∈ (1, ∞), then the range of the function σ

_{−r}is dense in the interval if and only if r ≤ η. We end with an open problem.