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
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $31$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $31$