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

Source: IMO Longlist 1989, Problem 79

September 18, 2008

inductionalgebra unsolvedalgebra

Problem Statement

Given two natural numbers

w

and

n,

the tower of

n

w's

is the natural number

T_n(w)

defined by

T_n(w) = w^{w^{\cdots^{w}}},

with

n

w's

on the right side. More precisely,

T_1(w) = w

and

T_{n+1}(w) = w^{T_n(w)}.

For example,

T_3(2) = 2^{2^2} = 16,

T_4(2) = 2^{16} = 65536,

and

T_2(3) = 3^3 = 27.

Find the smallest tower of

3's

that exceeds the tower of

1989

2's.

In other words, find the smallest value of

n

such that

T_n(3) > T_{1989}(2).

Justify your answer.

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