Square of Small Repunit is Palindromic
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Theorem
The squares of repunits with up to $9$ digits are palindromic.
Proof
| \(\ds 1^2\) | \(=\) | \(\ds 1\) | ||||||||||||
| \(\ds 11^2\) | \(=\) | \(\ds 121\) | ||||||||||||
| \(\ds 111^2\) | \(=\) | \(\ds 12 \, 321\) | ||||||||||||
| \(\ds 1111^2\) | \(=\) | \(\ds 1 \, 234 \, 321\) | ||||||||||||
| \(\ds 11 \, 111^2\) | \(=\) | \(\ds 123 \, 454 \, 321\) | ||||||||||||
| \(\ds 111 \, 111^2\) | \(=\) | \(\ds 12 \, 345 \, 654 \, 321\) | ||||||||||||
| \(\ds 1 \, 111 \, 111^2\) | \(=\) | \(\ds 1 \, 234 \, 567 \, 654 \, 321\) | ||||||||||||
| \(\ds 11 \, 111 \, 111^2\) | \(=\) | \(\ds 123 \, 456 \, 787 \, 654 \, 321\) | ||||||||||||
| \(\ds 111 \, 111 \, 111^2\) | \(=\) | \(\ds 12 \, 345 \, 678 \, 987 \, 654 \, 321\) |
but:
- $1 \, 111 \, 111 \, 111^2 = 1 \, 234 \, 567 \, 900 \, 987 \, 654 \, 321$
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1,111,111,111,111,111,111$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1,111,111,111,111,111,111$