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

\(\displaystyle 8\) \(=\) \(\displaystyle 2^3\)
\(\displaystyle 9\) \(=\) \(\displaystyle 3^2\)


\(\displaystyle 288\) \(=\) \(\displaystyle 2^5 \times 3^2\)
\(\displaystyle 289\) \(=\) \(\displaystyle 17^2\)


\(\displaystyle 675\) \(=\) \(\displaystyle 3^3 \times 5^2\)
\(\displaystyle 676\) \(=\) \(\displaystyle 2^2 \times 13^2\)


\(\displaystyle 9800\) \(=\) \(\displaystyle 2^3 \times 5^2 \times 7^2\)
\(\displaystyle 9801\) \(=\) \(\displaystyle 3^4 \times 11^2\)


\(\displaystyle 332 \, 928\) \(=\) \(\displaystyle 2^7 \times 3^2 \times 17^2\)
\(\displaystyle 332 \, 929\) \(=\) \(\displaystyle 577^2\)

$\blacksquare$


Sources