Problem #407: Let $w(n)$ count the number of solutions...

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Progress

A conjecture originally due to Newman.

This is true, and was proved by Evertse, Györy, Stewart, and Tijdeman [EGST88].

Quantitative bounds were provided by Tijdeman and Wang [TiWa88], who proved that (if $w(n)$ only counts distinct solutions, where we call two solutions distinct if the sets $\{2^a,3^b,2^{c}3^d\}$ are distinct) then $w(n) \leq 4$ for all large $n$.

This was made effective by Bajpai and Bennett [BaBe24], who proved that $w(n)\leq 4$ if $n\geq 131082$ and $w(n)\leq 9$ for all $n$. (The largest $n$ for which $w(n)=9$ is $299$.) There are infinitely many $n$ with $w(n)=4$, given by the identities
\begin{align*}
2^{a-1}+3^b+2^{a-1}3^0
&= 2^{a-2}+3^b+2^{a-2}3^1\\
&=2^a+3^{b-1}+2\cdot 3^{b-1}\\
&=2^a+3^{b-2}+2^33^{b-2}.
\end{align*}