On the distance between products of consecutive Fibonacci numbers and powers of Fibonacci numbers
Let \((F_n)_{n \geq 0}\) be the Fibonacci sequence given by \(F_0=0, F_1=1\) and \(F_{n+2}=F_{n+1}+F_n\) for all \(n \geq 0\). In the paper under review the authors study a lower bound on the quantity \[ | F_n \cdots F_{n+k-1}-F_m^l | \] where \(k \geq 1, l \geq 1, n \geq 3, m\geq 3\). The main result of the paper is the following { Theorem 1.} Let \((k,l,m,n)\) be integers with the properties that \(k \geq 1, l \geq 1, n \geq 3\) and \(m\geq 3\). Put \(X:=\max\{k,l,m,n\}\). Then the inequality \[ | F_n \cdots F_{n+k-1}-F_m^l | >10^{-3/2}X^{1/{40}} \] holds always except when either \(l=k=1\) and \(m=n(=X)\) or \((k,l,m,n)=(1,3,3,6)\), for which the left-hand side of the above equation is 0. As a consequence of Theorem 1 the authors derive the following numerical corollary. Corollary 1. The largest solution of the inequality \[ | F_n \cdots F_{n+k-1}-F_m^l | \leq 100 \] with \(k \geq 1, l \geq 1, n \geq 3, m\geq 3\) and \(m \neq n\) when \(k=l=1\) is \[ | F_9 \cdots F_{13}-F_{11}^5|=89. \] Here, by the largest solution we mean the solution with the maximal value of \(\max\{F_n \cdots F_{n+k-1}, F_m^l\}\) among all the possible solutions. This maximal value equals \(5 584 059 449\). For the proof of Theorem 1, the authors combine Baker's method with some known facts about Fibonacci numbers and with some deep, earlier results, such as \textit{R. D. Carmichael}'s Primitive Divisor Theorem [Ann. Math. (2) 15, 30--48 (1913; JFM 44.0216.01), Ann. Math. (2) 15, 49--70 (1914; JFM 45.1259.10)], a result of \textit{Y. Bugeaud} et al. [Ann. Math. (2) 163, No. 3, 969--1018 (2006; Zbl 1113.11021)] on Fibonacci and Lucas perfect powers and a result of \textit{F. Luca} and \textit{T. N. Shorey} [J. Number Theory 114, No. 2, 298--311 (2005; Zbl 1081.11023)] concerning Diophantine equations with products of consecutive terms in Lucas sequences. For the proof of Corollary 1, the initial upper bound for \(X=\max\{k,l,m,n\}\) is \(X<1.2\cdot 10^{100}\), which is a consequence of Theorem 1 and the fact that \(| a| \leq 100\). Then the authors derive a sharp upper bound for \(k\) (i.e. \(k \leq 2075\)). Finally, they apply the LLL algorithm to reduce the bound on \(X\), completing the proof of Corollary 1.
application of linear forms in logarithms of algebraic numbers
scientific article; zbMATH DE number 6121468