Squares whose Digits form Consecutive Integers
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Theorem
The sequence of integers whose squares have a decimal representation consisting of the concatenation of $2$ consecutive integers, either increasing or decreasing begins:
- $91, 428, 573, 727, 846, 7810, 9079, 9901, 36 \, 365, 63 \, 636, 326 \, 734, 673 \, 267, 733 \, 674, \ldots$
This sequence can be divided into two subsequences:
Those where the consecutive integers are increasing:
- $428, 573, 727, 846, 7810, 36 \, 365, 63 \, 636, 326 \, 734, 673 \, 267, \ldots$
This sequence is A030467 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Those where the consecutive integers are decreasing:
- $91, 9079, 9901, 733 \, 674, 999 \, 001, 88 \, 225 \, 295, \ldots$
This sequence is A054216 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
We have:
\(\ds 91^2\) | \(=\) | \(\ds 8281\) | ||||||||||||
\(\ds 428^2\) | \(=\) | \(\ds 183 \, 184\) | ||||||||||||
\(\ds 573^2\) | \(=\) | \(\ds 328 \, 329\) | ||||||||||||
\(\ds 727^2\) | \(=\) | \(\ds 528 \, 529\) | ||||||||||||
\(\ds 846^2\) | \(=\) | \(\ds 715 \, 716\) | ||||||||||||
\(\ds 7810^2\) | \(=\) | \(\ds 6099 \, 6100\) | ||||||||||||
\(\ds 9079^2\) | \(=\) | \(\ds 82 \, 428 \, 241\) | ||||||||||||
\(\ds 9901^2\) | \(=\) | \(\ds 98 \, 029 \, 801\) | ||||||||||||
\(\ds 36 \, 365^2\) | \(=\) | \(\ds 13224 \, 13225\) | ||||||||||||
\(\ds 63 \, 636^2\) | \(=\) | \(\ds 40495 \, 40496\) | ||||||||||||
\(\ds 326 \, 734^2\) | \(=\) | \(\ds 106755 \, 106756\) |
They can be determined by inspection.
$\blacksquare$
Also see
- Squares whose Digits form Consecutive Increasing Integers
- Squares whose Digits form Consecutive Decreasing Integers
Sources
- 1960: Maurice Kraitchik: Mathematical Recreations (2nd ed.)
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $60,996,100$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $60,996,100$