**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

*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

**On sums of multiple squares**

*Original research paper. Pages 9—15*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 165 Kb) | Abstract

_{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

^{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

**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

**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

_{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

*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

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

*Book review. Page 72*

Anthony Shannon

Full paper (PDF, 101 Kb)