Algebraic divisibility sequences over function fields

Abstract

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.

28 pages

References in corpus (8)

• Primitive divisors of Lucas and Lehmer sequences
• Primitive divisors of elliptic divisibility sequences
• The elliptic curve discrete logarithm problem and equivalent hard problems for elliptic divisibility sequences
• Terms in elliptic divisibility sequences divisible by their indices
• Elliptic nets and elliptic curves
• Polynomial Zsigmondy theorems
• The uniform primality conjecture for elliptic curves
• Primitive Divisors of Certain Elliptic Divisibility Sequences

Cited by in corpus (5)

• Polynomial Zsigmondy theorems
• Common valuations of division polynomials
• Primitive divisors of sequences associated to elliptic curves over function fields
• Remarks on a theorem of Silverman
• Zsigmondy's theorem and primitive divisors of the Lucas and Lehmer sequences in polynomial rings