Three Tri-Automorphic Numbers for each Number of Digits
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Theorem
Let $d \in \Z_{>0}$ be a (strictly) positive integer.
Then there exist exactly $3$ tri-automorphic numbers with exactly $d$ digits.
These tri-automorphic numbers all end in $2$, $5$ or $7$.
Proof
Let $n$ be a tri-automorphic number with $d$ digits.
Let $n = 10 a + b$.
Then:
- $3 n^2 = 300a^2 + 60 a b + 3 b^2$
As $n$ is tri-automorphic, we have:
- $(1): \quad 300 a^2 + 60 a b + 3 b^2 = 1000 z + 100 y + 10 a + b$
and:
- $(2): \quad 3 b^2 - b = 10 x$
where $x$ is an integer.
This condition is only satisfied by $b = 2$, $b = 5$, or $b = 7$
This theorem requires a proof. In particular: Guess: Try proving for $n = 10 a + b$ and then by induction. 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. |
Substituting $b = 2$ in equation $(1)$:
- $a = 9$
Substituting $b = 5$ in equation $(1)$:
- $a = 7$
Substituting $b = 7$ in equation $(1)$:
- $a = 6$
$\blacksquare$
Examples
Tri-Automorphic Numbers with $4$ Digits
The $3$ tri-automorphic numbers with $4$ digits are:
\(\text {(6667)}: \quad\) | \(\ds 3 \times 6667^2\) | \(=\) | \(\ds 133 \, 34 \mathbf {6 \, 667}\) | |||||||||||
\(\text {(6875)}: \quad\) | \(\ds 3 \times 6875^2\) | \(=\) | \(\ds 141 \, 79 \mathbf {6 \, 875}\) | |||||||||||
\(\text {(9792)}: \quad\) | \(\ds 3 \times 9792^2\) | \(=\) | \(\ds 287 \, 64 \mathbf {9 \, 792}\) |
Tri-Automorphic Numbers with $10$ Digits
The $3$ tri-automorphic numbers with $10$ digits are:
\(\text {(6 666 666 667)}: \quad\) | \(\ds 3 \times 6 \, 666 \, 666 \, 667^2\) | \(=\) | \(\ds 133 \, 333 \, 333 \, 34 \mathbf {6 \, 666 \, 666 \, 667}\) | |||||||||||
\(\text {(7 262 369 792)}: \quad\) | \(\ds 3 \times 7 \, 262 \, 369 \, 792^2\) | \(=\) | \(\ds 158 \, 226 \, 044 \, 98 \mathbf {7 \, 262 \, 369 \, 792}\) | |||||||||||
\(\text {(9 404 296 875)}: \quad\) | \(\ds 3 \times 9 \, 404 \, 296 \, 875^2\) | \(=\) | \(\ds 265 \, 322 \, 399 \, 13 \mathbf {9 \, 404 \, 296 \, 875}\) |
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $6667$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $6667$