Primitive Divisors in Divisibility Sequences

Joseph Silverman, Brown University
A divisibility sequence is a sequence of integers D_n with the property that if m divides n, then D_m divides D_n. There are many interesting examples of such sequences arising from linear recurrences and from elliptic curves. A primitive divisor of D_n is a prime p that divides D_n, but such that p divides no earlier D_k in the sequence. A fundamental problem is to show that all but finitely many terms of a given (divisibility) sequence have a primitive divisor, and then to give uniform bounds for the largest n, or for the number of n, such that D_n does not have a primitive divisor. We discuss classical and recent work on these problems, especially for elliptic divisibility sequences and for divisibility sequences arising in dynamical systems.