# Volume 10, 2004, Number 4

**Volume 10** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**A Fermatian Staudt—Clausen Theorem**

*Original research paper. Pages 89—99*

A. Shannon

Full paper (PDF, 120 Kb) | Abstract

This paper looks at the Staudt—Clausen theorem within the framework of various generalization of the Bernoulli numbers. The historical background to the problem is reviewed, and a solution to a problem of Morgan Ward is put forward. Generalized Hurwitz series are utilised in the development of the results.

**An extension of Euler’s prime-generating function**

*Original research paper. Pages 100—105*

J. Leyendekkers and A. Shannon

Full paper (PDF, 90 Kb) | Abstract

Using integer structure, six simple functions are obtained to give values for *x* that result in composite *N* in Euler’s prime generating function *N* = *x*^{2} + *x *+ *p*; the remaining values for *x* yield primes. In 0 ≤ *x* ≤ 500, with *p* = 41, there are 314 values for *x* which generate primes, the formation of which follows an orderly pattern based on integer structure. All primes can be generated from *N* = 6*r* ± 1, with specific values of *r* being rejected, in an analogous manner to the *x* values.

**On two Smarandache’s problems**

*Original research paper. Pages 106—112*

Krassimir T. Atanassov and Mladen V. Vassilev-Missana

Full paper (PDF, 1893 Kb)

**Volume 10** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4