Perfect powers in the summatory function of the power tower (Q719144)

The main result of this paper is: let \((a_n)\) for \(n\geq1\) be the sequence given by \(a_1 =1\) and \(a_n =n^{a_{n-1}}\) for \(n\geq2\), then the only solution of the equation \[ a_1 +\cdots+a_n =m^h \] is in positive integers \(h>1\), \(m\) and \(n\) is \(m=n=1\). Notice that \(a_2=2\), \(a_3=9\), \(a_4=262144\), \(a_5>10^{183230}\), \dots\ is a sequence that increases extremely fast. This explains partly the result. The proof uses in particular Archimedean and \(p\)-adic Baker's theory about linear forms in logarithms.