Fibonacci numbers at most one away from a perfect power
In 2006, the first, third and fourth author of the present paper proved [Ann. Math. (2) 163, No. 3, 969--1018 (2006; Zbl 1113.11021)] that the only Fibonacci numbers that are perfect powers different from 0 and 1, are \(F_6=8\) and \(F_{12}=144\); and the only Lucas number that is a perfect power different from 1, is \(L_3=4\). We remind here that the initial terms of the Fibonacci sequence are \(F_0=0\), \(F_1=1\) and those of the Lucas sequence are \(L_0=2\), \(L_1=1\); and both sequences satisfy the second-order recurrence \(u_{n+2}=u_{n+1}+u_n\). In this nicely written paper the authors sketch briefly the proof of the above theorem, by successfully giving a feeling of how the so called \textit{modular approach} to exponential Diophantine equations (with its origins in the proof of \textit{Fermat's Last Theorem} by A.~Wiles) is combined with the lower bounds for linear forms in two or three logarithms of algebraic numbers. Moreover, combining the above strong theorem with only elementary (but not straightforward) arguments they prove that the only solutions \((n,y,p)\) to the equation \(F_n\pm 1=y^p\) with \(p\geq 2\) are furnished by the relations: \(F_0+1=1\), \(F_4+1=2^2\), \(F_6+1=3^2\), \(F_1-1=F_2-1=0\), \(F_3-1=1\) and \(F_5-1=2^2\). In their proof the authors take advantage of the fact that \(F_n+1\) is factored in the form \(F_aL_b\) for appropriate indices \(a,b\) depending on the class of \(n\bmod 4\).
modular approach to Diophantine equations
linear forms in two or three logarithms of algebraic numbers
scientific article; zbMATH DE number 5362007
scientific article