Common values of linear recurrences related to Shank's simplest cubics (Q6633559)

Let \( A,B,C \) be integers, not all zeros and let \( F(u,n)=F(A,B,C,u,n) \) be the linear recurrence 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 \( S_u(X)=X^3-(u-1)X^2-(u+2)X-1 \), where \( u\in \mathbb{Z} \). In the paper under review, in their main result, the authors prove that there exists an effectively computable constant \( c \) depending only on \( L=\max\{\vert A\vert, \vert B \vert, \vert C\vert\} \) such that if the Diophantine equation \N\[\N\vert F(A,B,C,u,n)\vert =\vert F(A,B,C,u,m)\vert\tag{1}\N\]\Nholds for some integers \( u,n,m \) with \( n\ne m \), then \( \vert n\vert, \vert m\vert < c \). Furthermore, for choices \( (A,B,C)\in \{(0,0,1), (1,-1,1)\} \), the authors solve the equation (1) completely. Finally, they also give an outlook on the Diophantine equation \( F(A,B,C,u,n)\vert =\vert F(A,B,C,v,m) \) for some fixed integers \( n,m \). The proofs of their results heavily rely clever combination of techniques in number theory and on the properties of the Shanks polynomials and the Shanks sequences, results on linear forms in logarithms of algebraic numbers as well as reduction techniques involving the theory of continued fractions. Some computations are done with the aid of computer algebra packages in \texttt{SageMath} and \texttt{Magma}.