examples of Kaprekar numbers

Take the integer 142857 in base 10. Its square is 20408122449, an 11-digit number, but for the sake of example let’s think of it as 020408122449. If we split it into two 6-digit numbers, 020408 and 122449, and add them up, we get 142857 back. Thus 142857 is a Kaprekar number in base 10.

The Mersenne numbers have plenty of base-dependent properties in binary, and one of them is that they are all Kaprekar numbers. For example, 127, which is 1111111 in binary. Its square is 11111100000001, and sure enough, 1111110 + 0000001 = 1111111.

It has been mentioned that $b^x-1$ is a Kaprekar number in base $b$. Take 999 for example. Its square is 998001, and it’s obvious that 998 + 001 = 999. From these observations we can generalize that ${(b^x-1)}^2=(b^x(b^x-1)-1)+1$.

Title	examples of Kaprekar numbers
Canonical name	ExamplesOfKaprekarNumbers
Date of creation	2013-03-22 16:02:00
Last modified on	2013-03-22 16:02:00
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	5
Author	PrimeFan (13766)
Entry type	Example
Classification	msc 11A63