6174 Explained

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:

1. Take any four-digit number, using at least two different digits (leading zeros are allowed).
2. Arrange the digits in descending and then in ascending order to get two four-digit numbers, adding leading zeros if necessary.
3. Subtract the smaller number from the bigger number.
4. 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]

Other "Kaprekar's constants"

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

Cryptography

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

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

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

• 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:
  • 18 + 18 + 18 = 5832 + 324 + 18 = 6174, and coincidentally, 6 + 1 + 7 + 4 = 18.
• The sum of squares of the prime factors of 6174 is a square:
  • 2 + 3 + 3 + 7 + 7 + 7 = 4 + 9 + 9 + 49 + 49 + 49 = 169 = 13

External links

• Web site: Bowley. Roger. 6174 is Kaprekar's Constant. Numberphile. 5 December 2011 . Brady Haran. University of Nottingham.
• 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

Notes and References

1. Web site: Nishiyama . Yutaka . Yutaka Nishiyama . March 2006 . Mysterious number 6174 . Plus Magazine.
2. Kaprekar DR . 1955 . An Interesting Property of the Number 6174 . . 15 . 244–245.
3. Kaprekar DR . 1980 . On Kaprekar Numbers . Journal of Recreational Mathematics . 13 . 2 . 81–82.
4. [Kaprekar's routine#CITEREFHanover2017|Hanover 2017]
5. Web site: Kaprekar's Iterations and Numbers . 2022-09-21 . www.cut-the-knot.org.
6. https://testbook.com/maths/kaprekars-constant#:~:text=Cryptography%3A%20Kaprekar's%20Constant%20is%20used,used%20to%20find%20prime%20numbers.