Padua and Pisa are exponentially far apart
The sequence of Fibonacci (or Leonardo of Pisa) numbers is given by \(F_0=0\), \(F_1=1\) and \(F_{n+1}=F_n+F_{n-1}\) for \(n\geq 1\). The Padovan numbers (named after Richard Padovan) are defined by \(P_0=1, P_1=0, P_2=0\) and the ternary recurrence \(P_{n+1}=P_{n-1}+P_{n-2}\) for \(n\geq 2\). Using lower bounds for linear forms in three logarithms, due to Baker-Wüstholz and Voutier, the author shows that if \(F_n\not= P_m\), then \[ | P_m-F_n| \geq P_m\cdot m^{-c} \] with some explicit constant \(c\), which amounts to saying that Pisa and Padua are exponentially far apart. Using a computational technique introduced by \textit{A. Baker} and \textit{H. Davenport} [Q. J. Math., Oxf. (2) 20, 129-137 (1969; Zbl 0177.06802)], the author shows that if \(| P_m-F_n| \leq P_m^{1/2}\), then \(m\leq 29\) and \(n\leq 15\). The author also obtains some other results on Fibonacci and Padovan numbers.
scientific article; zbMATH DE number 1146208