Linear forms in Lucas numbers

Abstract

If (u m) m∈ N 0 denotes a Lucas sequence, i.e. a binary integer recurrence sequence with initial values u 0=0 and u 1=1, then the equation ku m = lu n with k,l∈ Z⧹{0} and max{ m, n}≥5 can be valid only for finitely many Lucas sequences with coprime roots and finitely many indices m,n∈ N , which can - both - be effectively bounded. This yields lower bounds for | ku m − lu n |. In the same way the equation u n = l can be considered, and this gives a partial answer to a conjecture of Beukers concerning multiplicities of binary recurrences. The proofs depend on estimates for linear forms in logarithms and on bounds for the solutions of equations in binary forms.