  The busy beaver value BB(n) is the maximum number of steps made by any
n-state, 2-symbol deterministic halting Turing machine starting on blank tape.
The busy beaver function $n \mapsto \text{BB}(n)$ is uncomputable and, from
below, only 4 of its values, BB(1) ... BB(4), are known to date. This leads one
to ask: from above, what is the smallest BB value that encodes a major
mathematical challenge? Knowing BB(4,888) has been shown by Yedidia and
Aaronson [28] to be at least as hard as solving Goldbach's conjecture, with a
subsequent improvement, as yet unpublished, to BB(27) [4,1]. We prove that
knowing BB(15) is at least as hard as solving the following Collatz-related
conjecture by Erd\H{o}s, open since 1979 [9]: for all n > 8 there is at least
one digit 2 in the base 3 representation of $2^n$. We do so by constructing an
explicit 15-state, 2-symbol Turing machine that halts if and only if the
conjecture is false. This 2-symbol Turing machine simulates a conceptually
simpler 5-state, 4-symbol machine which we construct first. This makes, to
date, BB(15) the smallest busy beaver value that is related to a natural open
problem in mathematics, bringing to light one of the many challenges underlying
the quest of knowing busy beaver values.
