A Lucas–Lehmer approach to generalised Lebesgue–Ramanujan–Nagell equations

We describe a computationally efficient approach to resolving equations of the form $$C_1x^2 + C_2 = y^n$$ C 1 x 2 + C 2 = y n in coprime integers, for fixed values of $$C_1$$ C 1 , $$C_2$$ C 2 subject to further conditions. We make use of a factorisation argument and the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier.

Primitive divisors of sequences associated to elliptic curves with complex multiplication

Let P and Q be two points on an elliptic curve defined over a number field K. For $$\alpha \in {\text {End}}(E)$$ α ∈ End ( E ) , define $$B_\alpha $$ B α to be the $$\mathcal {O}_K$$ O K -integral ideal generated by the denominator of $$x(\alpha (P)+Q)$$ x ( α ( P ) + Q ) . Let $$\mathcal {O}$$ O be a subring of $${\text {End}}(E)$$ End ( E ) , that is a Dedekind domain. We will study the sequence $$\{B_\alpha \}_{\alpha \in \mathcal {O}}$$ { B α } α ∈ O . We will show that, for all but finitely many $$\alpha \in \mathcal {O}$$ α ∈ O , the ideal $$B_\alpha $$ B α has a primitive divisor when P is a non-torsion point and there exist two endomorphisms $$g\ne 0$$ g ≠ 0 and f so that $$f(P)= g(Q)$$ f ( P ) = g ( Q ) . This is a generalization of previous results on elliptic divisibility sequences.

ALGEBRAIC DIVISIBILITY SEQUENCES OVER FUNCTION FIELDS

In this note we study the existence of primes and of primitive divisors in function field analogues of classical divisibility sequences. 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.

Primitive divisors in arithmetic dynamics

Let ϕ(z) ∈ (z) be a rational function of degree d ≥ 2 with ϕ(0) = 0 and such that ϕ does not vanish to order d at 0. Let α ∈ have infinite orbit under iteration of ϕ and write ϕn(α) = An/Bn as a fraction in lowest terms. We prove that for all but finitely many n ≥ 0, the numerator An has a primitive divisor, i.e., there is a prime p such that p | An and p ∤ Ai for all i < n. More generally, we prove an analogous result when ϕ is defined over a number field and 0 is a preperiodic point for ϕ.

Primitive Divisors on Twists of Fermat's Cubic

We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u3 + v3 = m, with m cube-free, all the terms beyond the first have a primitive divisor.

Primitive divisors of Lucas and Lehmer sequences, III

In this paper we prove that if n>30 030, then the nth element of any Lucas or Lehmer sequence has a primitive divisor.