Properties of Family of 333,667 and Related Numbers/Squares
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Theorem
This page reports on certain properties, difficult to classify, of the number $333 \, 667$, and patterns arising.
The square of any number consisting of:
- a string of $3$s
followed by:
- a string of $6$s
followed by:
- a single $7$
has its digits all in an increasing sequence:
\(\ds 333 \, 667^2\) | \(=\) | \(\ds 111 \, 333 \, 666 \, 889\) | ||||||||||||
\(\ds 33 \, 366 \, 667^2\) | \(=\) | \(\ds 1 \, 113 \, 334 \, 466 \, 688 \, 889\) |
The same is true of numbers of the following form:
\(\ds 16 \, 667^2\) | \(=\) | \(\ds 277 \, 788 \, 889\) | ||||||||||||
\(\ds 333 \, 334^2\) | \(=\) | \(\ds 111 \, 111 \, 555 \, 556\) | ||||||||||||
\(\ds 333 \, 335^2\) | \(=\) | \(\ds 111 \, 112 \, 222 \, 225\) |
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. |
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $333,667$
- Feb. 1987: Roger B. Nelsen and R. Glenn Powers: Solutions to Problem 1234 (Math. Mag. Vol. 60, no. 1: pp. 46 – 47) www.jstor.org/stable/2690137
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $333,667$