# Square Cullen Numbers

## Theorem

The numbers:

$1, 9, 25$

are Cullen numbers which are also square.

## Proof

We have:

 $\displaystyle 1$ $=$ $\displaystyle 0 \times 2^0 + 1$ $\displaystyle 9$ $=$ $\displaystyle 2 \times 2^2 + 1$ $\displaystyle 25$ $=$ $\displaystyle 3 \times 2^3 + 1$

$\blacksquare$