Powers of 2 which are Sum of Distinct Powers of 3

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Unproven Conjectures

The powers of $2$ which are sums of distinct powers of $3$ are:

\(\displaystyle 2^0\) \(=\) \(\, \displaystyle 1 \, \) \(\, \displaystyle =\, \) \(\displaystyle 3^0\)
\(\displaystyle 2^2\) \(=\) \(\, \displaystyle 4 \, \) \(\, \displaystyle =\, \) \(\displaystyle 3^0 + 3^1\)
\(\displaystyle 2^8\) \(=\) \(\, \displaystyle 256 \, \) \(\, \displaystyle =\, \) \(\displaystyle 3^0 + 3^1 + 3^2 + 3^5\)

It has been conjectured by Paul Erdős that there are no others.


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