Squares whose Digits form Consecutive Decreasing Integers
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Theorem
The sequence of integers whose squares have a decimal representation consisting of the concatenation of $2$ consecutive decreasing integers begins:
- $91, 9079, 9901, 733 \, 674, 999 \, 001, 88 \, 225 \, 295, 99 \, 990 \, 001, \ldots$
This sequence is A030467 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
We have:
\(\ds 91^2\) | \(=\) | \(\ds 8281\) | ||||||||||||
\(\ds 9079^2\) | \(=\) | \(\ds 82 \, 428 \, 241\) | ||||||||||||
\(\ds 9901^2\) | \(=\) | \(\ds 98 \, 029 \, 801\) | ||||||||||||
\(\ds 733 \, 674^2\) | \(=\) | \(\ds 538 \, 277 \, 538 \, 276\) | ||||||||||||
\(\ds 999 \, 001^2\) | \(=\) | \(\ds 998 \, 002 \, 998 \, 001\) |
They can be determined by inspection.
$\blacksquare$