Dattaraya Ramchandra Kaprekar

Dattaraya Ramchandra Kaprekar (1905-01-17 - 1986) was an Indian mathematician who discovered many interesting properties in number theory. He was born in the town Devlali, Maharashtra. Having never received any formal postgraduate training, for his entire career (1930-1962) he was a schoolteacher in the small town of Devlali in Maharashtra, India. Yet he became well known in recreational mathematics circles, and has a number, a constant, and a magic square named after him.

Kaprekar received his secondary school education in Thana and studied at Fergusson College in Pune. He attended the University of Bombay, receiving his bachelor"s degree in 1929. He published extensively, writing about such topics as recurring decimals, magic squares, and integers with special properties.

Working largely alone, Kaprekar discovered a number of results that have opened up important avenues of research in number theory. In addition to the Kaprekar constant and the Kaprekar number which were named after him, he also discovered the Self number or Devlali number, and also the important series called the Harshad number. He also constructed certain types of magic squares related to the Copernicus magic square.

Kaprekar"s famous works are below:
Kaprekar constant
The Kaprekar constant, named after him, is a fascinating general property of all number bases and may demonstrate some important but unknown theorem in number theory.The Kaprekar constant, or 6174 (1949). He showed that 6174 is reached in the limit as one repeatedly subtracts the highest and lowest numbers that can be constructed from four digits (not all the same). This series converges to 6174 in fewer than seven iterations.

Ex: lets take a non-zero 4 digit number 5, 5, 9, 2,
9552-2559=6993, 9963-3699=6264, 6642-2466=4176, 7641-1467=6174, 7641-1467=6174...this goes on.

Similarly the Kaprekar constant for 3 digits is 495.

Ex: lets take a non-zero 3 digit number 5, 5, 9,
955-559=396, 963-369=594, 954-459=495, 954-495=495...this goes on.

Kaprekar number
The Kaprekar number (also called Kaprekar series, based on the Kaprekar operation). This is a number with the interesting property that if it is squared, then two equal parts of this square also add up to the original number. This operation, of taking the last n digits of a square, and adding it to the number formed by the first (n-1) or n digits, is the Kaprekar operation.

Ex: an example 297
297^2 = 88,209 and 88 + 209 = 297!!!

So, just square the number, split it in half (leave the larger portion to the right) and add the two halves - if you get the original number back, then its a Kaprekar Number.

A few more examples of Kaprekar Numbers are - 9, 45, 297, 4879, 17344, 538461, ...

Devlali or Self number
In 1963, he also defined the property which has come to be known as self numbers, which are integers that cannot be generated by taking some other number and adding its own digits to it. For example, 21 is not a self number, since it can be generated from 15: 15 + 1 + 5 = 21. But 20 is a self number, since it cannot be generated from any other integer. He also gave a test for verifying this property in any number. These are sometimes referred to as Devlali numbers (after the town where he lived, Devlali,Maharashtra); though this appears to have been his preferred designation, the term self-number is more widespread. Sometimes these are also designated Colombian numbers after a later designation.

Harshad Number
He also discovered the Harshad numbers which he named harshad, meaning "giving joy" (Sanskrit harsha, joy +da taddhita pratyaya, causative); these have the property that they are divisible by the sum of their digits. Thus 12, which is divisible by 1 + 2 = 3, is a Harshad number. These were later also called Niven numbers after a 1997 lecture on these by the Canadian mathematician Ivan M. Niven. Numbers which are Harshad in all bases (only 1, 2, 4, and 6) are called all-Harshad numbers. Much work has been done on Harshad numbers, and their distribution, frequency, etc. are a matter of considerable interest in number theory today.

In 1927 he won the Wrangler R. P. Paranjpe Mathematical Prize for an original piece of work in mathematics.

Although he was largely unknown outside recreational mathematics circles in his lifetime, D. R. Kaprekar"s work and its impact on number theory has become widely recognized in recent years.

courtesy:internet