Volume 7, 2001, Number 4

Volume 7Number 1Number 2Number 3 ▷ Number 4

q-Bernoulli numbers and polynomials via an invariant p-adic q-integral in Zp
Original research paper. Pages 105—110
H.-K. Pak and S.-H. Rim
Full paper (PDF, 87 Kb) | Abstract

We define the q-Bernoulli numbers by using an p-adic q-integral due to T. Kim and investigate the properties of these numbers. In the final section, we will give the formula for sums of products of these numbers.

Intervals containing prime numbers
Original research paper. Pages 111—114
L. Panaitopol
Full paper (PDF, 122 Kb) | Abstract

For x > 0, let π(x) be the number of prime numbers not exceeding x. One shows that, for x 7, there exists at least one prime number between x and x + π(x), thus obtaining a result that is sharper than the one postulated by Bertrand.

The analysis of twin primes within Z6
Original research paper. Pages 115—124
J. Leyendekkers and A. Shannon
Full paper (PDF, 490 Kb) | Abstract

The modular ring Z6 defines integers via (6ri + (i – 3)) where i is the Class and r, the row when tabulated in an array. Since only Classes 26 and 46 (; contain odd primes, this modular ring is ideally suited for the analysis of twin primes. In considering a series of integers, a simple method is used to calculate rows (F rows) that do not contain twin primes. This allows the distribution of other primes to be found. Then, in considering the corresponding array of rows, elimination of the F rows yields the rows which contain twin primes. The calculations are facilitated by the use of the right-end digit (RED) technique.

Some properties of modified Lah numbers
Original research paper. Pages 125—131
A. Shannon
Full paper (PDF, 180 Kb) | Abstract

A modification of Lah numbers is suggested in this paper by defining them in relation to the rising factorial coefficients instead of the falling factorial coefficients. Some of their properties are then developed, particularly those in relation to Bernoulli and Stirling numbers and Laguerre polynomials. A partial recurrence relation for the modified Lah numbers is also studied.

How Fermat’s Great Theorem helps solving the Diophantine equation
12.x3φ(y).y2 = 3.(φ(y))3

Original research paper. Pages 132—134
M. Vassilev-Missana
Full paper (PDF, 102 Kb)

On one Jacobsthal’s inequality
Original research paper. Pages 135—136
K. Atanassov
Full paper (PDF, 45 Kb)

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