Volume 10, 2004, Number 4

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.

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.