Square of Repdigit Number consisting of Instances of 3

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Theorem

The following pattern holds:

\(\ds 3^2\) \(=\) \(\ds 09\)
\(\ds 0 + 9\) \(=\) \(\ds 9\)


\(\ds 33^2\) \(=\) \(\ds 1089\)
\(\ds 10 + 89\) \(=\) \(\ds 99\)


\(\ds 333^2\) \(=\) \(\ds 110 \, 889\)
\(\ds 110 + 889\) \(=\) \(\ds 999\)


\(\ds 3333^2\) \(=\) \(\ds 11 \, 108 \, 889\)
\(\ds 1110 + 8889\) \(=\) \(\ds 9999\)


\(\ds 33 \, 333^2\) \(=\) \(\ds 1 \, 111 \, 088 \, 889\)
\(\ds 11 \, 110 + 88 \, 889\) \(=\) \(\ds 99 \, 999\)

and so on.


Proof


This theorem requires a proof.
In particular: Simple but tedious.
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Sources

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