D. R. Kaprekar

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Dattaraya Ramchandra Kaprekar (17 January 1905 in Dahanu, Maharashtra - 1986) was an Indian mathematician who discovered many interesting properties in number theory. 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 [1].

Kaprekar received his secondary school education in Thana and studied at Fergusson College in Pune. In 1927 he won the Wrangler R. P. Paranjpe Mathematical Prize for an original piece of work in mathematics[2]. 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. In particular, the Kaprekar constant is named after him.

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[edit] Discoveries

Working largely alone, Kaprekar discovered a number of results in number theory and described various properties of numbers. In addition to the Kaprekar constant and the Kaprekar number which were named after him, he also described the Self number or Devlali number and the series called the Harshad numbers. He also constructed certain types of magic squares related to the Copernicus magic square[3].

[edit] Kaprekar constant

One of his most fascinating discoveries is the Kaprekar constant, or 6174 (1949)[4]. 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). Thus, starting with 1234, we have

4321 - 1234 = 3087,

then

8730 - 0378 = 8352, and
8532 - 2358 = 6174.

Repeating from now on leaves the same number (7641 - 1467 = 6174). This series converges to 6174 in at most seven iterations for all four-digit numbers.

A similar constant for 3 digits is 495[5]. However, in base 10 a single such constant only exists for numbers of 3 or 4 digits.

[edit] Kaprekar number

Another important contribution is known as the Kaprekar number[6] (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 (e.g. 45, since 452=2025, and 20+25=45. Also 9, 55, 99 etc. 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.

9^2 = 81 ... 8+1 = 9

45^2 = 2025 ... 20+25 = 45

55^2 = 3025 ... 30+25 = 55

99^2 = 9801 ... 98+01 = 99

703^2 = 494209 ... 494+209 = 703

999^2 = 998001 ... 998+001 = 999

2728^2 = 7441984 ... 744+1984 = 2728

4950^2 = 24502500 ... 2450+2500 = 4950

5050^2 = 25502500 ... 2550+2500 = 5050

7272^2 = 52881984 ... 5288+1984 = 7272

7777^2 = 60481729 ... 6048+1729 = 7777

9999^2 = 99980001 ... 9998+0001 = 9999

17344^2 = 300814336 ... 3008+14336 = 17344

22222^2 = 493817284 ... 4938+17284 = 22222

77778^2 = 6049417284 ... 60494+17284 = 77778

82656^2 = 6832014336 ... 68320+14336 = 82656

95121^2 = 9048004641 ... 90480+04641 = 95121

99999^2 = 9999800001 ... 99998+00001 = 99999

187110^2 = 35010152100 ... 35010+152100 = 187110

318682^2 = 101558217124 ... 101558+217124 = 318682

329967^2 = 108878221089 ... 108878+221089 = 329967

351352^2 = 123448227904 ... 123448+227904 = 351352

356643^2 = 127194229449 ... 127194+229449 = 356643

390313^2 = 152344237969 ... 152344+237969 = 390313

461539^2 = 213018248521 ... 213018+248521 = 461539

466830^2 = 217930248900 ... 217930+248900 = 466830

499500^2 = 249500250000 ... 249500+250000 = 499500

500500^2 = 250500250000 ... 250500+250000 = 500500

533170^2 = 284270248900 ... 284270+248900 = 533170

538461^2 = 289940248521 ... 289940+248521 = 538461

609687^2 = 371718237969 ... 371718+237969 = 609687

643357^2 = 413908229449 ... 413908+229449 = 643357

648648^2 = 420744227904 ... 420744+227904 = 648648

670033^2 = 448944221089 ... 448944+221089 = 670033

681318^2 = 464194217124 ... 464194+217124 = 681318

791505^2 = 626480165025 ... 626480+165025 = 791505

812890^2 = 660790152100 ... 660790+152100 = 812890

818181^2 = 669420148761 ... 669420+148761 = 818181

851851^2 = 725650126201 ... 725650+126201 = 851851

857143^2 = 734694122449 ... 734694+122449 = 857143

961038^2 = 923594037444 ... 923594+037444 = 961038

994708^2 = 989444005264 ... 989444+005264 = 994708

999999^2 = 999998000001 ... 999998+000001 = 999999

[edit] Devlali or Self number

In 1963, he also defined the property which has come to be known as self numbers[7], 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); though this appears to have been his preferred designation[7], the term self-number is more widespread. Sometimes these are also designated Colombian numbers after a later designation.

[edit] Harshad Number

He also described the Harshad numbers which he named harshad, meaning "giving joy" (Sanskrit harsha, joy +da taddhita pratyaya, causative); these are defined by 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.

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[1].

[edit] References

  1. ^ a b Kenneth H. Rosen (2005). Elementary Number Theory and its Applications. Addison Wesley. 
  2. ^ Dilip M. Salwi (2005-01-24). Dattaraya Ramchandra Kaprekar. Retrieved on 2007-11-30.
  3. ^ Kaprekar, D. R. (1974) The Copernicus Magic Square. Indian Journal Of History Of Science, Vol. 9, No. 1
  4. ^ Kaprekar, D. R., "Another Solitaire Game", Scripta Mathematica, vol 15, pp 244-245 (1949)
  5. ^ http://mathpoint.blogspot.com/2006/12/mysterious-6174-revisited.html An informal proof of the property for three digits.
  6. ^ Kaprekar Number - from Wolfram MathWorld
  7. ^ a b Kaprekar, D. R. The Mathematics of New Self-Numbers Devalali (1963): 19-20