On Prime-Detecting Sequences From Apéry's Recurrence Formulae for ζ(3) and ζ(2)

We consider the linear three-term recurrence formula

$\begin{displaymath}X_n = (34{(n-1)}^3 + 51{(n-1)}^2 + 27(n-1) +5) X_{n-1} - {(n-1)}^6 X_{n-2} \quad (n\geq 2) \end{displaymath}$

corresponding to Apéry's non-regular continued fraction for $\zeta(3)$ . It is shown that integer sequences ${(X_n)}_{n\geq 0}$ with $5X_0 \not= X_1$ satisfying the above relation are prime-detecting, i.e., $X_n \not\equiv 0 \,(\bmod \,n)$ if and only if $n$ is a prime not dividing $\vert 5X_0 - X_1\vert$ . Similar results are given for integer sequences satisfying the recurrence formula

$\begin{displaymath}X_n = (11{(x-1)}^2 + 11(x-1) + 3) X_{n-1} + {(n-1)}^4 X_{n-2} \quad (n\geq 2) \end{displaymath}$

corresponding to Apéry's non-regular continued fraction for $\zeta(2)$ and for sequences related to $\log 2$ .

Received July 25 2008; revised version received October 23 2008. Published in Journal of Integer Sequences, November 16 2008.

On Prime-Detecting Sequences From Apéry's Recurrence Formulae for ζ(3) and ζ(2)

Carsten Elsner Fachhochschule für die Wirtschaft University of Applied Sciences Freundallee 15 D-30173 Hannover Germany

Carsten Elsner
Fachhochschule für die Wirtschaft
University of Applied Sciences
Freundallee 15
D-30173 Hannover
Germany