On Prime-Detecting Sequences From Apery's Recurrence Formulae for (3) and (2)
Carsten Elsner · 2008
We consider the linear three-term recurrence formula Xn = (34(n − 1) 3 + 51(n − 1) 2 + 27(n − 1) + 5)Xn−1 − (n − 1) 6 Xn−2 (n ≥ 2) corresponding to Apery's non-regular continued fraction for �(3). It is shown that integer sequences (Xn) n≥0 with 5X0 6 X1 satisfying the above relation are prime- detecting, i.e., Xn 6≡ 0(modn) if and only if n is a prime not dividing |5X0 − X1|. Similar results are given for integer sequences satisfying the recurrence formula Xn = (11(x − 1) 2 + 11(x − 1) + 3)Xn−1 + (n − 1) 4 Xn−2 (n ≥ 2) corresponding to Apery's non-regular continued fraction for �(2) and for sequences related to log2.