# Volume 7, 2001, Number 1

Volume 7 ▶ Number 1 ▷ Number 2Number 3Number 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., ) 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

In what follows we shall study certain arithmetic functions with application to the study of some trigonometrical products.

Some new formulae for the twin primes counting function π2(n)
Original research paper. Pages 10–14
M. Vassilev-Missana
Full paper (PDF, 151 Kb) | Abstract

For every n > 1, let π2(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 π2(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

Let C = {Cn}n≥1 be an arbitrary increasing sequence of natural numbers. By πC(n) we denote the number of the terms of C being not greater than n (we agree that πC(0) = 0). In the first part of the paper we propose six different formulae for Cn (n = 1, 2, …), which depend on the numbers πC(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 pn, 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-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 h2n−1 using modular rings ℤ6 and ℤ4
Original research paper. Pages 21–28
J. Leyendekkers and A. Shannon
Full paper (PDF, 354 Kb) | Abstract

Twin primes of the form h2n−1 are analysed within the modular ring ℤ6. The values of h are odd and 3|h for the lowest valued twin prime, p2. The other values of h fall in either the second (26) or fourth (46) class of ℤ6, depending on the parity of n. Functional relationships are developed for the various hn and rows within ℤ6. All p2 fall in class 26 and the larger prime of the twin pair, p4, always falls in class 26. With n4 = 1 and n4 > 1, p2 falls in class one (14) of the modular ring ℤ4 and hence equals a unique set of squares (x2 + y2). Analysis of the distribution of the (x,y) pair reveals an interesting prime sequence related to the Fibonacci sequence.

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