Squares whose Digits form Consecutive Decreasing Integers

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.

$\blacksquare$

Also see

• Squares whose Digits form Consecutive Integers
• Squares whose Digits form Consecutive Increasing Integers