How many \(k\)-digit Fibonacci numbers do exist? (Q1203567)

The Fibonacci sequence is given by \(F_ 1=F_ 2=1\) and \(F_{n+1}=F_ n+ F_{n-1}\) for \(n\geq 2\). Let \(a(k)\) be the number of indices \(n\) such that \(F_ n\) has \(k\) decimal digits. Then \(a(k)\in\{4,5\}\) for \(k\geq 2\). Let \(A(N)\) be the number of \(k\) with \(2\leq k\leq N\) and \(a(k)=5\). Among other things, the author shows that \(A(N)=\alpha N+O(1)\) as \(N\to\infty\), where \(\alpha=\log 10/\log((1+ \sqrt{5})/2)- 4=0.78497\ldots\;\). His proof uses Baker's method on linear forms in logarithms.