Existence of primitive divisors of Lucas and Lehmer numbers

Abstract

We prove that for n b 30, every n-th Lucas and Lehmer number has a primitive divisor. This allows us to list all Lucas and Lehmer numbers without a primitive divisor.

On Generalized Lebesgue-Ramanujan-Nagell Equations

“…The key ingredient of this method is the powerful result of Bilu, Hanrot and Voutier [19] concerning primitive prime prime divisors of Lucas sequences. To illustrate the method we solve completely the equation Proof.…”

“…In this case the Primitive Divisor Theorem of Bilu, Hanrot and Voutier [19] can be applied very efficiently.…”

“…is a prime then L n has a primitive prime divisor except for finitely many pairs (α, β) which are explicitly determined in Table 1 of [19].…”

Balancing Diophantine triples with distance 1

“…Note that ε z = 1 admits B 2y−1 = B 2 y , which contradicts the Primitive Divisor Theorem (see [10]) since y is large enough. Supposing ε z = −1, we obtain ≤ α z+1 k +2(k+ )+2k+2 −4·0.983 .…”

Bennett’s Pillai theorem with fractional bases and negative exponents allowed

“…Now we can apply Lemma 3. In this appendix we give a short elementary proof of Proposition 4.1 below, which is proven in [11] using lower bounds on linear forms in logarithms, a deep result of Bennett and Skinner [5], and the well known result of Bilu, Hanrot, and Voutier [6]. .…”