Square Cullen Numbers

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Theorem

The numbers:

$1, 9, 25$

are Cullen numbers which are also square.


Proof

We have:

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

$\blacksquare$


Sources