Sum of Successive Powers in 2 ways

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.

$\blacksquare$