October 16, 2025Open Access

The order of appearance of the product of the first and second Lucas numbers

HLHongjian LiUniversity of Science and Technology HXHuiming XiaoHangzhou Academy of Agricultural Sciences PYPingzhi YuanSouth China Normal University

Key Points

The order of appearance of an integer in the first lucas sequence is defined with respect to gcd properties.
Explicit formulae for τ(U_m V_n) and other combinations are provided when parameters meet specific conditions.
The study generalizes previous results by examining interactions between the first and second lucas sequences.
It emphasizes the role of positive integers and their gcd relations in determining properties of lucas sequences.

Abstract

Let a and b be relatively prime integers. Then the first Lucas sequence (Uₙ) ₍₀ and the second Lucas sequence (Vₙ) ₍₀ are defined respectively by U₍+₂=aU₍+₁+bU₍, \, U₀=0, \, U₁=1 and V₍+₂=aV₍+₁+bV₍, \, V₀=2, \, V₁=a, where n0. Let m be an integer with (m, \, b) =1. Then the smallest positive integer k satisfying m Uₖ is called the order of appearance of m in the first Lucas sequence (Uₙ) ₍₀, denoted by τ (m), i. e. , τ (m): =\k1: m Uₖ\. When a>0 and Δ=a²+4b>0, we give explicit formulae for τ (Uₘ Vₙ), τ (Uₘ Uₙ), τ (Vₘ Vₙ) and τ (UₙU₍+₏U₍+₂₏), thus generalizing the results of Irmak and Ray.

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