The number 6174 is known as Kaprekar's constant [1] [2] [3] after the Indian mathematician D. R. Kaprekar. This number is renowned for the following rule:
The above process, known as Kaprekar's routine, will always reach its fixed point, 6174, in at most 7 iterations. [4] Once 6174 is reached, the process will continue yielding 7641 – 1467 = 6174. For example, choose 1459:
The only four-digit numbers for which Kaprekar's routine does not reach 6174 are repdigits such as 1111, which give the result 0000 after a single iteration. All other four-digit numbers eventually reach 6174 if leading zeros are used to keep the number of digits at 4. For numbers with three identical digits and a fourth digit that is one higher or lower (such as 2111), it is essential to treat 3-digit numbers with a leading zero; for example: 2111 – 1112 = 0999; 9990 – 999 = 8991; 9981 – 1899 = 8082; 8820 – 288 = 8532; 8532 – 2358 = 6174. [5]
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Cardinal | six thousand one hundred seventy-four | |||
Ordinal | 6174th (six thousand one hundred seventy-fourth) | |||
Factorization | 2 × 32 × 73 | |||
Divisors | 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 49, 63, 98, 126, 147, 294, 343, 441, 686, 882, 1029, 2058, 3087, 6174 | |||
Greek numeral | ,ϚΡΟΔ´ | |||
Roman numeral | VMCLXXIV, or VICLXXIV | |||
Binary | 11000000111102 | |||
Ternary | 221102003 | |||
Senary | 443306 | |||
Octal | 140368 | |||
Duodecimal | 36A612 | |||
Hexadecimal | 181E16 |
There can be analogous fixed points for digit lengths other than four; for instance, if we use 3-digit numbers, then most sequences (i.e., other than repdigits such as 111) will terminate in the value 495 in at most 6 iterations. Sometimes these numbers (495, 6174, and their counterparts in other digit lengths or in bases other than 10) are called "Peyush constants" named after Peyush Dixit who solved this routine as a part of his IMO 2000 (International Mathematical Olympiad, Year 2000) thesis. [6]
In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for "repeated unit" and was coined in 1966 by Albert H. Beiler in his book Recreations in the Theory of Numbers.
In mathematics, a natural number in a given number base is a -Kaprekar number if the representation of its square in that base can be split into two parts, where the second part has digits, that add up to the original number. For example, in base 10, 45 is a 2-Kaprekar number, because 45² = 2025, and 20 + 25 = 45. The numbers are named after D. R. Kaprekar.
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In number theory, Kaprekar's routine is an iterative algorithm named after its inventor, Indian mathematician D. R. Kaprekar. Each iteration starts with a number, sorts the digits into descending and ascending order, and calculates the difference between the two new numbers.
Dattatreya Ramchandra Kaprekar was an Indian recreational mathematician who described several classes of natural numbers including the Kaprekar, harshad and self numbers and discovered the Kaprekar's constant, named after him. Despite having no formal postgraduate training and working as a schoolteacher, he published extensively and became well known in recreational mathematics circles.
In number theory, a Dudeney number in a given number base is a natural number equal to the perfect cube of another natural number such that the digit sum of the first natural number is equal to the second. The name derives from Henry Dudeney, who noted the existence of these numbers in one of his puzzles, Root Extraction, where a professor in retirement at Colney Hatch postulates this as a general method for root extraction.
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I myself once learned 380 digits of π, when I was a crazy high-school kid. My never-attained ambition was to reach the spot, 762 digits out in the decimal expansion, where it goes "999999", so that I could recite it out loud, come to those six 9's, and then impishly say, "and so on!"
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In number theory, a perfect digit-to-digit invariant is a natural number in a given number base that is equal to the sum of its digits each raised to the power of itself. An example in base 10 is 3435, because . The term "Munchausen number" was coined by Dutch mathematician and software engineer Daan van Berkel in 2009, as this evokes the story of Baron Munchausen raising himself up by his own ponytail because each digit is raised to the power of itself.
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