Algebraic divisibility sequences over function fields

Patrick Ingram, Valéry Mahé, Joseph H. Silverman, Katherine E. Stange, Marco Streng

Published 2011-05-27, updated 2011-10-27Version 2

We study the existence of primes and of primitive divisors in classical divisibility sequences defined over function fields. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.