Erdős Problem #407

Let \(w(n)\) count the number of solutions to \[n=2^a+3^b+2^c3^d\] with \(a,b,c,d\geq 0\) integers. Is it true that \(w(n)\) is bounded by some absolute constant? A conjecture originally due to Newman. This is true, and was proved by Evertse, Győry, 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 \(299\).)