Squares whose Digits form Consecutive Integers

From ProofWiki
Jump to navigation Jump to search

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


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Squares_whose_Digits_form_Consecutive_Integers&oldid=384841"