# Consecutive Powerful Numbers

## Theorem

The following pairs are of consecutive positive integers both of which are powerful:

$\left({8, 9}\right), \left({288, 289}\right), \left({675, 676}\right), \left({9800, 9801}\right), \left({332 \, 928, 332 \, 929}\right), \ldots$

## Proof

 $\displaystyle 8$ $=$ $\displaystyle 2^3$ $\displaystyle 9$ $=$ $\displaystyle 3^2$

 $\displaystyle 288$ $=$ $\displaystyle 2^5 \times 3^2$ $\displaystyle 289$ $=$ $\displaystyle 17^2$

 $\displaystyle 675$ $=$ $\displaystyle 3^3 \times 5^2$ $\displaystyle 676$ $=$ $\displaystyle 2^2 \times 13^2$

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

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

$\blacksquare$