Consecutive Powerful Numbers
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Theorem
The following pairs are of consecutive positive integers both of which are powerful:
- $\left({8, 9}\right), \left({288, 289}\right), \left({675, 676}\right), \left({9800, 9801}\right), \left({332 \, 928, 332 \, 929}\right), \ldots$
This sequence is A060355 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
\(\ds 8\) | \(=\) | \(\ds 2^3\) | ||||||||||||
\(\ds 9\) | \(=\) | \(\ds 3^2\) |
\(\ds 288\) | \(=\) | \(\ds 2^5 \times 3^2\) | ||||||||||||
\(\ds 289\) | \(=\) | \(\ds 17^2\) |
\(\ds 675\) | \(=\) | \(\ds 3^3 \times 5^2\) | ||||||||||||
\(\ds 676\) | \(=\) | \(\ds 2^2 \times 13^2\) |
\(\ds 9800\) | \(=\) | \(\ds 2^3 \times 5^2 \times 7^2\) | ||||||||||||
\(\ds 9801\) | \(=\) | \(\ds 3^4 \times 11^2\) |
\(\ds 332 \, 928\) | \(=\) | \(\ds 2^7 \times 3^2 \times 17^2\) | ||||||||||||
\(\ds 332 \, 929\) | \(=\) | \(\ds 577^2\) |
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $288$