Sum of Successive Powers in 2 ways

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Theorem

$31$ and $8191$ can be expressed as the sum of successive powers starting from $1$ in in $2$ different ways.


Proof

\(\ds 31\) \(=\) \(\ds 1 + 5 + 5^2\)
\(\ds \) \(=\) \(\ds 1 + 2 + 2^2 + 2^3 + 2^4\)


\(\ds 8191\) \(=\) \(\ds 1 + 90 + 90^2\)
\(\ds \) \(=\) \(\ds 1 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 + 2^9 + 2^{10}+ 2^{11}+ 2^{12}\)

These are the only two examples known.

$\blacksquare$


Sources