The set of primes dividing the Lucas numbers has density 2∕3

Jeffrey C. Lagarias

Vol. 118 (1985), No. 2, 449–461

DOI: 10.2140/pjm.1985.118.449

Abstract

The Lucas numbers L_n are defined by L₀ = 2, L₁ = 1 and the recurrence L_n = L_n−1 + L_n−2. The set of primes S_L = {p : p divides L_n for some n} has density 2/3. Similar density results are proved for sets of primes S_U = {p : p divides U_n for some n} for certain other special second-order linear recurrences {U_n}. The proofs use a method of Hasse.

Mathematical Subject Classification 2000

Primary: 11B05

Secondary: 11B39

Milestones

Received: 12 September 1984

Revised: 28 September 1984

Published: 1 June 1985