On the Shorey-Tijdeman Diophantine equation involving terms of Lucas sequences (Q2076509)

Let \(r\) be a positive integer, \(\phi =\frac12 (\sqrt 5+1)\) be the smallest possible of \(\alpha\) and \(\mathbf{U}=\{ U_n\}_{n\geq 0}\) be the Lucas sequence given by \(U_0=0\), \(U_1=1\) and \(U_{n+2}=rU_{n+1}+U_n\) for all \(n\geq 0\). Let \(\alpha\) and \(\beta\) be the roots of the characteristic polynomial \(X^2-rX-1\). Let \(A\), \(B\), \(C\) and \(D\) be integers such that \\ \(AU_n+BU_m=CU_{n_1}+DU_{m_1}\), \(n>m\geq 0\), \(n_1>m_1\geq 0\) and \(AU_n\not=CU_{n_1}\). (1)\\ Set \(X=\max(\mid A\mid , \mid B\mid ,\mid C\mid , \mid D\mid)\). Then the authors prove the following two theorems. \\ Theorem 1: Let \(AB\not=0\) and (1) holds. Then \(r<14X\). \\ Theorem 2: Relabeling the variables \((n, m, n_1, m_1)\) to \((n_1, n_2, n_3, n_4)\) such that \(n_1\geq n_2 \geq n_3 \geq n_4\). If \(n_1=n_2\) we rewrite (1) as \((A-C)U_n+BU_m=DU_{m_1}\) and change \((A, B, C, D)\) to \((A-C, B, D, 0)\). Thus, \(n_1>n_2\). Furthermore we change the sign of some of the coefficients \((A, B, C, D)\) so that (1) becomes \\ \(A_1U_{n_1}+A_2U_{n_2}+A_3U_{n_3}+A_4U_{n_4}=0\). \ (2)\\ Assume \(r\leq 14X\). Then the solutions of (2) are of two types. \\ 1. Finitely many solutions with \(n_4\leq \frac{\log (8X)}{\log \phi}\), \(n_3\leq \frac{\log (400X^3)}{\log \phi}\), \(n_2\leq \frac{\log (32000X^5)}{\log \phi}\), \(n_1\leq \frac{\log (640000X^6)}{\log \phi}\). \\ 2. There are one of the two forms. \\ 2a. \((n_1, n_2, n_3, n_4)=(n_3+j, n_3+i, n_3, 0)\), \(i\leq 2\frac{\log (8X)}{\log \phi}\), \(j\leq \frac{\log (500X^3)}{\log \phi}\) and \(\alpha\) is a root of \(A_1X^i+A_2X^j+A_3=0\) or of the form\\ 2b. \((n_1, n_2, n_3, n_4)=(n_4+k, n_4+j, n_4+i, n_4)\), \(i\leq \frac{\log (50X^2)}{\log \phi}\), \(j\leq \frac{\log (1600X^3)}{\log \phi}\), \(j\leq \frac{\log (25000X^4)}{\log \phi}\) and \(\alpha\) is a root of \(A_1X^k+A_2X^j+A_3X^i+A_4=0\).