**Volume 18** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**On certain bounds and limits for prime numbers**

*Original research paper. Pages 1—5*

József Sándor

Full paper (PDF, 165 Kb) | Abstract

We will consider various limits and inequalities connected with the *n*-th prime number.

**A note on the area and volume of right-angled triangles with integer sides**

*Original research paper. Pages 6—8*

Winston Buckley

Full paper (PDF, 117 Kb) | Abstract

We show that the area of every right–angled triangle with integer sides is a multiple of 6, and its volume, the product of the lengths of all three sides, is a multiple of 60.

**On sums of multiple squares**

*Original research paper. Pages 9—15*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 165 Kb) | Abstract

The structural and other characteristics of the Hoppenot multiple square equation are analysed in the context of the modular ring Z_{4}. This equation yields a left-hand-side and a right-hand-side sum equal to P_{n} (24T_{n} + 1) in which P_{n}, T_{n} represent the pyramidal and triangular numbers, respectively. This sum always has 5 as a factor. Integer structure analysis is also used to solve some related problems.

**Conductors for sets of large integer squares**

*Original research paper. Pages 16—21*

Ken Dutch and Christy Rickett

Full paper (PDF, 134 Kb) | Abstract

We calculate the Frobenius conductor for the infinite set {n^{2}, (n + 1)^{2}, …} through n = 200, demonstrate that the conductor’s growth rate as function of n is o(n^{2+ε}) for any positive ε, and calculate specific numerical bounds for several ε > 0.0145.

**Double inequalities on means via quadrature formula**

*Original research paper. Pages 22—28*

K. M. Nagaraja and P. Siva Kota Reddy

Full paper (PDF, 195 Kb) | Abstract

In this paper, using Simpson’s quadrature formula and Jensen inequality for convex function, we obtained some double inequalities among various means.

**Patterns related to the Smarandache circular sequence primality problem**

*Original research paper. Pages 29—48*

Marco Ripà and Emanuele Dalmasso

Full paper (PDF, 736 Kb) | Abstract

In this paper, we show the internal relations among the elements of the circular sequence (1, 12, 21, 123, 231, 312, 1234, 3412, …). We illustrate one method to minimize the number of the “candidate prime numbers” up to a given term of the sequence. So, having chosen a particular prime divisor, it is possible to analyze only a fixed number of the smallest terms belonging to a given range, thus providing the distribution of that prime factor in a larger set of elements. Finally, we combine these results with another one, also expanding the study to a few new integer sequences related to the circular one.

**The number of primitive Pythagorean triples in a given interval**

*Original research paper. Pages 49—57*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 184 Kb) | Abstract

Integer structure analysis within the framework of modular rings is used to show that the formation of primitive (or “reduced”) Pythagorean triples depends on certain characteristics with these rings. Only integers in the Class ̅1_{4} of the modular ring Z_{4} can produce primitive Pythagorean triples. Of these, a prime produces only one primitive Pythagorean triple, while composites produce the same number of primitive Pythagorean triples as their factors, provided the factors are square-free or are not elements of ̅3_{4} . Class ̅1_{4} integers were converted to the equivalent Z_{6} classes in order to isolate those divisible by 3. The numbers of primitive Pythagorean triples in various ranges were estimated and compared with the elder Lehmer’s estimates. The results provide a neat link between the number of primitive Pythagorean triples and the number of primes in the given interval. It was also shown why the major component of a primitive Pythagorean triple is the only component which cannot have 3 as a factor.

**Note on ***φ*, *ψ* and *σ-*functions. Part 5

*Original research paper. Pages 58—62*

Krassimir Atanassov

Full paper (PDF, 142 Kb) | Abstract

Inequalities connecting *φ*, *ψ* and *σ-*functions are formulated and proved.

**A short note on the Inclusion-Exclusion principle: A modification with applications**

*Original research paper. Pages 63—71*

Acquaah Peter

Full paper (PDF, 205 Kb) | Abstract

The Inclusion-Exclusion (I.E.) principle is an important counting concept in combinatorics. It is also very important in the study of the distribution of prime numbers. In this paper, we introduce an equivalent – and in some cases a relatively easier to apply – form of the concept. We also provide some applications.

*Elliptic Tales: Curves, Counting and Number Theory *by Avner Ash and Robert Gross

*Book review. Page 72*

Anthony Shannon

Full paper (PDF, 101 Kb)

**Volume 18** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4