Properties of Family of 333,667 and Related Numbers/Product with Certain Repetitive Numbers

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Theorem

This page reports on certain properties, difficult to classify, of the number $333 \, 667$, and patterns arising.


\(\ds 333 \, 667 \times 296\) \(=\) \(\ds 98 \, 765 \, 432\)
\(\ds 33 \, 336 \, 667 \times 2996\) \(=\) \(\ds 99 \, 876 \, 654 \, 332\)
\(\ds 3 \, 333 \, 366 \, 667 \times 29 \, 996\) \(=\) \(\ds 99 \, 987 \, 666 \, 543 \, 332\)


\(\ds 333 \, 667 \times 1113\) \(=\) \(\ds 371 \, 371 \, 371\)
\(\ds 33 \, 336 \, 667 \times 11 \, 133\) \(=\) \(\ds 371 \, 137 \, 113 \, 711\)
\(\ds 3 \, 333 \, 366 \, 667 \times 111 \, 333\) \(=\) \(\ds 371 \, 113 \, 711 \, 137 \, 111\)


\(\ds 333 \, 667 \times 2223\) \(=\) \(\ds 741 \, 741 \, 741\)
\(\ds 33 \, 336 \, 667 \times 22 \, 233\) \(=\) \(\ds 741 \, 174 \, 117 \, 411\)
\(\ds 3 \, 333 \, 366 \, 667 \times 222 \, 333\) \(=\) \(\ds 741 \, 117 \, 411 \, 174 \, 111\)


Proof


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In particular: Straightforward but tedious exercise in induction and heavy algebra.
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Historical Note

This result was reported by David Wells in his Curious and Interesting Numbers of $1986$ as having appeared in articles by H. Grunbaum in Volumes $18$ and $21$ of Scripta Mathematica.

However, this has not been corroborated.


Sources

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