6174
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:
- Take any four-digit number, using at least two different digits (leading zeros are allowed).
- Arrange the digits in descending and then in ascending order to get two four-digit numbers, adding leading zeros if necessary.
- Subtract the smaller number from the bigger number.
- Go back to step 2 and repeat.
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:
- 9541 – 1459 = 8082
- 8820 – 0288 = 8532
- 8532 – 2358 = 6174
- 7641 – 1467 = 6174
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]
| ||||
|---|---|---|---|---|
| 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 | |||
Other "Kaprekar's constants"
[edit]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 "Kaprekar constants".
Applications
[edit]Cryptography
[edit]Kaprekar's constant is often used in cryptography for the purpose of generating random numbers. Kaprekar's routine offers a way to arrive to completely random numbers which can be used for decryption and encryption. This technique is also often used to generate random prime numbers.
Convergence analysis
[edit]In numerical analysis, Kaprekar's constant can be used to analyze the convergence of a variety numerical methods. Numerical methods are used in engineering, various forms of calculus, coding, and many other mathematical and scientific fields.
Recursion theory
[edit]The properties of Kaprekar's routine allows for the study of recursive functions, ones which repeat previous values and generating sequences based on these values. Kaprekar's routine is a recursive arithmetic sequence, so it helps study the properties of recursive functions. [6]
Other properties
[edit]- 6174 is a 7-smooth number, i.e. none of its prime factors are greater than 7.
- 6174 can be written as the sum of the first three powers of 18:
- 183 + 182 + 181 = 5832 + 324 + 18 = 6174, and coincidentally, 6 + 1 + 7 + 4 = 18.
- The sum of squares of the prime factors of 6174 is a square:
- 22 + 32 + 32 + 72 + 72 + 72 = 4 + 9 + 9 + 49 + 49 + 49 = 169 = 132
References
[edit]- ^ Nishiyama, Yutaka (March 2006). "Mysterious number 6174". Plus Magazine.
- ^ Kaprekar DR (1955). "An Interesting Property of the Number 6174". Scripta Mathematica. 15: 244–245.
- ^ Kaprekar DR (1980). "On Kaprekar Numbers". Journal of Recreational Mathematics. 13 (2): 81–82.
- ^ Hanover 2017, p. 1, Overview.
- ^ "Kaprekar's Iterations and Numbers". www.cut-the-knot.org. Retrieved 2022-09-21.
- ^ https://testbook.com/maths/kaprekars-constant#:~:text=Cryptography%3A%20Kaprekar's%20Constant%20is%20used,used%20to%20find%20prime%20numbers.
External links
[edit]
- Bowley, Roger (5 December 2011). "6174 is Kaprekar's Constant". Numberphile. University of Nottingham: Brady Haran.
- Sample (Perl) code to walk any four-digit number to Kaprekar's Constant
- Sample (Python) code to walk any four-digit number to Kaprekar's Constant
- Sample (C) code to walk the first 10000 numbers and their steps to Kaprekar's Constant