LINEAR-FORMS IN LUCAS-NUMBERS

If (u(m))m is-an-element-of N0 denotes a Lucas sequence, i.e. a binary integer recurrence sequence with initial values u0 = 0 and u1 = 1, then the equation ku(m) = lu(n) with k,l is-an-element-of Z \ {0} and max{m,n} greater-than-or-equal-to 5 can be valid only for finitely many Lucas sequences with coprime roots and finitely many indices m,n is-an-element-of 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.