View source for 6174

{{Short description|Natural number and Kaprekar's constant}}
The number '''6174''' is known as '''Kaprekar's constant'''<ref>{{Cite web |last=Nishiyama |first=Yutaka |authorlink=Yutaka Nishiyama |date=March 2006 |title=Mysterious number 6174 |url=http://plus.maths.org/issue38/features/nishiyama/index.html |website=[[Plus Magazine]]}}</ref><ref name="Kaprekar1955">{{cite journal |author=Kaprekar DR |year=1955 |title=An Interesting Property of the Number 6174 |journal=[[Scripta Mathematica]] |volume=15 |pages=244–245}}</ref><ref name="Kaprekar1980">{{cite journal |author=Kaprekar DR |year=1980 |title=On Kaprekar Numbers |journal=Journal of Recreational Mathematics |volume=13 |issue=2 |pages=81–82}}</ref> after the [[India]]n [[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 (mathematics)|fixed point]], 6174, in at most 7 iterations.<ref>[[Kaprekar's routine#CITEREFHanover2017|Hanover 2017]], p. 1, Overview.</ref> Once 6174 is reached, the process will continue yielding 7641 – 1467 = 6174. For example, choose 1459:
{{startplainlist|indent=1}}
*9541 – 1459 = 8082
*8820 – 0288 = 8532
*8532 – 2358 = 6174
*7641 – 1467 = '''6174'''
{{endplainlist}}
The only four-digit numbers for which Kaprekar's routine does not reach 6174 are [[repdigit]]s such as 1111, which give the result [[0 (number)|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.<ref>{{Cite web |title=Kaprekar's Iterations and Numbers |url=https://www.cut-the-knot.org/Curriculum/Arithmetic/Kaprekar.shtml |access-date=2022-09-21 |website=www.cut-the-knot.org}}</ref>{{Infobox number
|number= 6174
| roman = {{overline|V}}MCLXXIV, or {{overline|VI}}CLXXIV
|divisor=1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 49, 63, 98, 126, 147, 294, 343, 441, 686, 882, 1029, 2058, 3087, 6174}}
==Other "Kaprekar's constants"==
{{Main|Kaprekar's routine#Definition and properties}}
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 (number)|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 [[Encryption|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 [[rate of convergence|convergence of a variety numerical methods]]. Numerical methods are used in [[engineering]], various forms of [[calculus]], [[computer programming|coding]], and many other mathematical and scientific fields.
=== Recursion theory ===
The properties of Kaprekar's routine allows for the study of [[General recursive function|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. <ref>https://testbook.com/maths/kaprekars-constant#:~:text=Cryptography%3A%20Kaprekar's%20Constant%20is%20used,used%20to%20find%20prime%20numbers.</ref>
==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{{sup|3}} + 18{{sup|2}} + 18{{sup|1}} = 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{{sup|2}} + 3{{sup|2}} + 3{{sup|2}} + 7{{sup|2}} + 7{{sup|2}} + 7{{sup|2}} = 4 + 9 + 9 + 49 + 49 + 49 = 169 = 13{{sup|2}}
==References==
{{Reflist}}
==External links==
{{commons category}}
* {{cite web|last=Bowley|first=Roger|title=6174 is Kaprekar's Constant|url=https://www.youtube.com/watch?v=d8TRcZklX_Q|work=Numberphile|date=5 December 2011 |publisher=[[Brady Haran]]|location=[[University of Nottingham]]}}
* [http://www.panix.com/~baldwin/kaprekar.pl Sample (Perl) code to walk any four-digit number to Kaprekar's Constant]
* [http://www.panix.com/~baldwin/kaprekar.py Sample (Python) code to walk any four-digit number to Kaprekar's Constant]
* [https://gist.github.com/dakonr/10865d113dd2ca6a7dd3d3fcdb169c13#file-kaprekar_constant-c Sample (C) code to walk the first 10000 numbers and their steps to Kaprekar's Constant]
[[Category:Arithmetic dynamics]]
[[Category:Mathematical constants|Kaprekar's constant]]
[[Category:Integers]]