Volume 7, 2001, Number 4

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.

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 modular ring Z₆ defines integers via (6r_i + (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.

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.