**Volume 7** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**Remarks on φ, σ, ψ and ρ functions**

*Original research paper. Pages 1–3*

K. Atanassov

Full paper (PDF, 105 Kb) | Abstract

*φ*and

*σ*functions (see, e.g., [1]) are two of the most important arithmetic functions. They have a lot of very interesting properties. Some of them will be discussed below.

**On a even perfect and superperfect numbers**

*Original research paper. Pages 4–5*

J. Sándor

Full paper (PDF, 58 Kb)

**On arithmetic functions and a trigonometrical product**

*Original research paper. Pages 6–9*

J. Sándor and L. Tóth

Full paper (PDF, 140 Kb) | Abstract

**Some new formulae for the twin primes counting function π_{2}(n)**

*Original research paper. Pages 10–14*

M. Vassilev-Missana

Full paper (PDF, 151 Kb) | Abstract

*n*> 1, let

*π*

_{2}(

*n*) denote the number of primes

*p*such that

*p*≤

*n*and

*p*+ 2 is also a prime. In the present paper some new formulae for

*π*

_{2}(

*n*) are proposed.

**Three formulae for n-th prime and six for n-th term of twin primes**

*Original research paper. Pages 15–20*

M. Vassilev-Missana

Full paper (PDF, 191 Kb) | Abstract

*C*= {

*C*}

_{n}_{n≥1}be an arbitrary increasing sequence of natural numbers. By

*π*(

_{C}*n*) we denote the number of the terms of

*C*being not greater than

*n*(we agree that

*π*(0) = 0). In the first part of the paper we propose six different formulae for

_{C}*C*(

_{n}*n*= 1, 2, …), which depend on the numbers

*π*(

_{C}*k*) (

*k*= 0, 1, 2, …). Using these formulae, in the second part of the paper we obtain three different explicit formulae for the

*n*-th prime

*p*, which are the first main result of the present research. In the third part of the paper, using the formulae from the first part, we propose six explicit formulae for the

_{n}*n*-th term of the sequence of twin primes: 3, 5, 7, 11, 13, 17, 19,… – the second main result of the paper. The last three of them are main ones for the twin primes.

**An analysis of twin primes h2^{n}−1 using modular rings ℤ_{6} and ℤ_{4}**

*Original research paper. Pages 21–28*

J. Leyendekkers and A. Shannon

Full paper (PDF, 354 Kb) | Abstract

*h*2

^{n}−1 are analysed within the modular ring ℤ

_{6}. The values of

*h*are odd and 3|

*h*for the lowest valued twin prime,

*p*

_{2}. The other values of

*h*fall in either the second (2

_{6}) or fourth (4

_{6}) class of ℤ

_{6}, depending on the parity of

*n*. Functional relationships are developed for the various

*h*,

*n*and rows within ℤ

_{6}. All

*p*

_{2}fall in class 2

_{6}and the larger prime of the twin pair,

*p*

_{4}, always falls in class 2

_{6}. With

*n*

_{4}= 1 and

*n*

_{4}> 1,

*p*

_{2}falls in class one (1

_{4}) of the modular ring ℤ

_{4}and hence equals a unique set of squares (

*x*

^{2}+

*y*

^{2}). Analysis of the distribution of the (

*x*,

*y*) pair reveals an interesting prime sequence related to the Fibonacci sequence.