What is 111111111 squared

Number Curiosities - Numbers with interesting behavior

The squares of the number sequence 1, 11, 111, ... always result in mirror numbers, also called palindromes (running backwards). They are the mathematical counterpart to the word palindromes such as ANNA or LAGERREGAL. The scheme ends - unfortunately - with the 9th link of the single successes.

1 · 1 = 1
11 · 11 = 121
111 · 111 = 12321
1111 · 1111 = 1234321
11111 · 11111 = 123454321
... = ...
111111111 · 111111111 = 12345678987654321

Various pairs of numbers mirrored to the vertical axis can be found, the results of which are also mirrored to the vertical after squaring. Here is a small selection:

1132 = 12769  ||  3112 = 96721
12122 = 1468944  ||  21212 = 4498641
122 = 144  ||  212 = 441
132 = 169  ||  312 = 961
1122 = 12544  ||  2112 = 44521


The palindromes presented above have something very special to offer: If they are multiplied by the sum of their individual digits, the result can be written as a sequence of special square numbers.

The result of the first five terms can still be confirmed with most pocket calculators. From 6666662 the (shortened) exponential notation is output. Anyone who doubts the exact continuity has to multiply by hand. A more convenient solution is to use a programming language that calculates with at least 18-digit integers in the output.


2592 = 25 · 92

This equation is meant to be unique.
Who comes up with something like that? Mathematicians are very special individuals!