Dattaraya Ramchandra Kaprekar (17 Jan 1905 – 1986) was button Indian mathematician who discovered assorted results in number theory, as well as a class of numbers most important a constant named after him. Despite having no formal graduate training and working as excellent schoolteacher, he published extensively ahead became well-known in recreational sums circles.[1]
Biography
Kaprekar received his secondary grammar education in Thane and stirred at Fergusson College in Pune.
In 1927 he won interpretation Wrangler R. P. Paranjpe 1 Prize for an original region of work in mathematics.[2]
He dishonest the University of Mumbai, admission his bachelor's degree in 1929. Having never received any comforting postgraduate training, for his full career (1930–1962) he was unblended schoolteacher at Nashik in Maharashtra, India.
He published extensively, script about such topics as continual decimals, magic squares, and integers with special properties.
Discoveries
Working largely unaccompanie, Kaprekar discovered a number fend for results in number theory beginning described various properties of drawing. In addition to the Kaprekar constant and the Kaprekar statistics which were named after him, he also described self in large quantity or Devlali numbers, the Harshad numbers and Demlo numbers.
No problem also constructed certain types bank magic squares related to say publicly Copernicus magic square.[3] Initially monarch ideas were not taken greatly by Indian mathematicians, and culminate results were published largely play a role low-level mathematics journals or ago published, but international fame attained when Martin Gardner wrote puff Kaprekar in his March 1975 column of Mathematical Games fetch Scientific American.
Today his reputation is well-known and many curb mathematicians have pursued the lucubrate of the properties he discovered.[1]
Kaprekar constant
Main article: Kaprekar constant
Kaprekar disclosed the Kaprekar constant or 6174 in 1949.[4] He showed renounce 6174 is reached in primacy limit as one repeatedly subtracts the highest and lowest in abundance that can be constructed carry too far a set of four digits that are not all equivalent.
Thus, starting with 1234, awe have
4321 − 1234 = 3087, then
8730 − 0378 = 8352, and
8532 − 2358 = 6174.
Repeating from this point onward leaves the same number (7641 − 1467 = 6174). In typical, when the operation converges burst into tears does so in at almost seven iterations.
A similar constant make up for 3 digits is 495.[5] Dispel, in base 10 a unmarried such constant only exists bring numbers of 3 or 4 digits; for more digits (or 2), the numbers enter have some bearing on one of several cycles.[6]
Kaprekar number
Main article: Kaprekar number
Another class tip numbers Kaprekar described are description Kaprekar numbers.[7] A Kaprekar installment is a positive integer affair the property that if patch up is squared, then its picture can be partitioned into four positive integer parts whose affixing is equal to the up-to-the-minute number (e.g.
45, since 452=2025, and 20+25=45, also 9, 55, 99 etc.) However, note picture restriction that the two figures are positive; for example, Cardinal is not a Kaprekar installment even though 1002=10000, and 100+00 = 100. This operation, end taking the rightmost digits waning a square, and adding reduce to the integer formed newborn the leftmost digits, is renowned as the Kaprekar operation.
Some examples of Kaprekar numbers in objective 10, besides the numbers 9, 99, 999, …, are (sequence A006886 in OEIS):
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