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
This theorem requires a proof. In particular: Straightforward but tedious exercise in induction and heavy algebra. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
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
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $333,667$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $333,667$