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## Schanuel’s conjecture and the Mandelbrot Competition

A student I’m tutoring was working unsuccessfully on the following problem from the 2011 Mandelbrot Competition:

Let $a, b$ be positive integers such that $\log_a b = (\log 23)(\log_6 7) + \log_2 3 + \log_6 7$. Find the minimum value of $ab$.

After some tinkering, I concluded that the problem as stated has no solution. I am now almost certain it was printed incorrectly: $\log 23$ should be replaced by $\log_2 3$, and then we can solve the problem as follows:

$\log_a b + 1 = (\log_2 3 + 1)(\log_6 7 + 1) = \log_2 6 \log_6 42 = \log_2 42$.

It follows that $\log_a b = \log_2 21$. Since $a, b$ are positive integers we must have $a \ge 2$, and then it follows that the smallest solution occurs when $a = 2, b = 21$. But what I’d like to discuss, briefly, is the argument showing that the misprinted problem has no solution.

For two categories $C, D$ let $D^C$ denote the functor category, whose objects are functors $C \to D$ and whose morphisms are natural transformations. For $C$ a locally small category, the Yoneda embedding is the functor $C \to \text{Set}^{C^{op}}$ sending an object $x \in C$ to the contravariant functor $\text{Hom}(-, x)$ and sending a morphism $x \to y$ to the natural transformation $\text{Hom}(-, x) \to \text{Hom}(-, y)$ given by composition. The goal of the next few posts is to discuss some standard properties of this embedding and try to gain some intuition about it.