Monday, August 12, 2019


      D.R.KAPRELAR DISCOVERIES
Kaprekar constant
In 1949, Kaprekar discovered an interesting property of the number 6174, which was subsequently named the Kaprekar constant. He showed that 6174 is reached in the limit as one repeatedly subtracts the highest and lowest numbers that can be constructed from a set of four digits that are not all identical. Thus, starting with 1234, we 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 general, when the operation converges it does so in at most seven iterations.
A similar constant for 3 digits is 495. However, in base 10 a single such constant only exists for numbers of 3 or 4 digits; for other digit lengths or bases other than 10, the Kaprekar's routine algorithm described above may in general terminate in multiple different constants or repeated cycles, depending on the starting value.
Kaprekar number
Another class of numbers Kaprekar described are the Kaprekar numbers. A Kaprekar number is a positive integer with the property that if it is squared, then its representation can be partitioned into two positive integer parts whose sum is equal to the original number (e.g. 45, since 452=2025, and 20+25=45, also 9, 55, 99 etc.) However, note the restriction that the two numbers are positive; for example, 100 is not a Kaprekar number even though 1002=10000, and 100+00 = 100. This operation, of taking the rightmost digits of a square, and adding it to the integer formed by the leftmost digits, is known as the Kaprekar operation.
Some examples of Kaprekar numbers in base 10, besides the numbers 9, 99, 999, …, are (sequence A006886 in the OEIS):
Number
Square
Decomposition
703
703² = 494209
494+209 = 703
2728
2728² = 7441984
744+1984 = 2728
5292
5292² = 28005264
28+005264 = 5292

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