Let (a_n)_{n\ge1} be a
sequence of nonzero integers. The
Zsigmondy set associated to the sequence is the
set :\mathcal{Z}(a_n) = \{n \ge 1 : a_n \text{ has no primitive prime divisors}\}. i.e., the set of indices n such that every prime dividing a_n also divides some a_m for some m . Thus Zsigmondy's theorem implies that \mathcal{Z}(a^n-b^n)\subset\{1,2,6\}, and
Carmichael's theorem says that the Zsigmondy set of the
Fibonacci sequence is \{1,2,6,12\}, and that of the
Pell sequence is \{1\}. In 2001 Bilu, Hanrot, and Voutier proved that in general, if (a_n)_{n\ge1} is a
Lucas sequence or a
Lehmer sequence, then \mathcal{Z}(a_n) \subseteq \{ 1 \le n \le 30 \} (see , there are only 13 such ns, namely 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 18, 30). Lucas and Lehmer sequences are examples of
divisibility sequences. It is also known that if (W_n)_{n\ge1} is an
elliptic divisibility sequence, then its Zsigmondy set \mathcal{Z}(W_n) is
finite. However, the result is ineffective in the sense that the proof does not give an explicit
upper bound for the largest element in \mathcal{Z}(W_n), although it is possible to give an effective upper bound for the number of elements in \mathcal{Z}(W_n). ==See also==