Inspired by Patrick Corn and Myself

Number Theory Level pending

Let \( a_{n+1} = 2^{a_n} \) and \( b_{n+1} = 3^{b_n} \) both for \( n \ge 1.\)

If \( a_1 = 2 \) and \(b_1 = 3\), then find the minimum \(l\) and maximum \(k\) such that

\(a_{n+k}<b_{n}<a_{n+l}\) for all \(n \geq 2\)

Input your answer as \(k+l\).


Inspiration

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