Volume 10, 2004, Number 4

Volume 10Number 1Number 2Number 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 = x2 + 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 = 6r ± 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 10Number 1Number 2Number 3 ▷ Number 4

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