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a_m= 2^{2^{,,,{^{2}}}}}, A_{km} (x)=2^{2^{,,,^{x^{a_m}}}}} B_k(y)=4^{,,,^{4^y}}}

Source: Austrian Federal Competition For Advanced Students 2009, Part 2, p1

August 31, 2019
algebranumber theoryexponential

Problem Statement

If x,y,K,m∈Nx,y,K,m \in N, let us define:
am=22,,,2k twosa_m= \underset{k \, twos}{2^{2^{,,,{^{2}}}}}, Akm(x)=22,,,xamk twosA_{km} (x)= \underset{k \, twos}{ 2^{2^{,,,^{x^{a_m}}}}}, Bk(y)=444,,,4ym foursB_k(y)= \underset{m \, fours}{4^{4^{4^{,,,^{4^y}}}}},
Determine all pairs (x,y)(x,y) of non-negative integers, dependent on k>0k>0, such that Akm(x)=Bk(y)A_{km} (x)=B_k(y)
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