Square Cullen Numbers

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Theorem

The numbers:

$1, 9, 25$

are Cullen numbers which are also square.


Proof

We have:

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

$\blacksquare$


Sources