Pair of Consecutive Powerful Numbers whose First is Odd

Theorem

The only known pair of consecutive integers which are both powerful numbers such that the first of the pair is odd is:

$\tuple {675, 676}$

Proof

By investigation:

\(\ds 675\)

\(=\)

\(\ds 3^3 \times 5^2\)

\(\ds 676\)

\(=\)

\(\ds 2^2 \times 13^2\)

That there are no smaller ones can be determined again by investigation.

$\blacksquare$

Sources

• Oct. 1970: S.W. Golomb: Powerful Numbers (Amer. Math. Monthly Vol. 77, no. 8: pp. 848 – 852)  www.jstor.org/stable/2317020

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $675$