Common values of linear recurrences related to Shank's simplest cubics

Abstract

Let A,B,C∈Z not all zeroes and let F(u,n)=F(A,B,C,u,n) be the linear recursive sequence, which is defined by the initial terms F(u,0)=A,F(u,1)=B,F(u,2)=C and whose characteristic polynomial is Daniel Shanks simplest cubic Su(X)=X3−(u−1)X2−(u+2)X−1,u∈Z. We prove that there exists an effectively computable constant c depending only on L=max⁡{|A|,|B|,|C|} such that if |F(A,B,C,u,n)|=|F(A,B,C,u,m)| holds for some integers u,n,m with n≠m then |n|,|m|<c. For the choices (A,B,C)∈{(0,0,1),(1,−1,1)} we solve the above equations completely. At the end we give an outlook to the equation F(0,0,1,u,n)=F(0,0,1,v,m) for some fixed integers n,m.