Volume 18, 2012, Number 1

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

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.

The structural and other characteristics of the Hoppenot multiple square equation are analysed in the context of the modular ring Z₄. 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.

We calculate the Frobenius conductor for the infinite set {n², (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.

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

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.

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 ̅1₄ of the modular ring Z₄ 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 ̅3₄ . Class ̅1₄ integers were converted to the equivalent Z₆ 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.

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

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.