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79

Part of 1989 IMO Longlists

Problems(1)

Power tower s.t. T_n(3) > T_{1989}(2)

Source: IMO Longlist 1989, Problem 79

9/18/2008
Given two natural numbers w w and n, n, the tower of n n w′s w's is the natural number Tn(w) T_n(w) defined by Tn(w)=ww⋯w, T_n(w) = w^{w^{\cdots^{w}}}, with n n w′s w's on the right side. More precisely, T1(w)=w T_1(w) = w and Tn+1(w)=wTn(w). T_{n+1}(w) = w^{T_n(w)}. For example, T3(2)=222=16, T_3(2) = 2^{2^2} = 16, T4(2)=216=65536, T_4(2) = 2^{16} = 65536, and T2(3)=33=27. T_2(3) = 3^3 = 27. Find the smallest tower of 3′s 3's that exceeds the tower of 1989 1989 2′s. 2's. In other words, find the smallest value of n n such that Tn(3)>T1989(2). T_n(3) > T_{1989}(2). Justify your answer.
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