1. Start with a four-digit number whose digits are not all equal.
2. Arrange the digits in ascending, say B, and descending order, say A.
3. Find A-B.
4. Go to step-2.
The above process terminates on the number 6174 after seven or fewer steps.
Example:
5063 --> 6530-0356=6174 --> 7641-1467=6174
The Indian mathematician Dattathreya Ramachandra Kaprekar (1905-1988) published the above property of the number 6174 in 1955 [#]. There are other interesting Kaprekar numbers too [#].
[#] Kaprekar, D.R., An Interesting Property of the Number 6174, Scripta Mathematica 15, 1955, pp. 244-245.
6 comments:
This is different.
The one which I mentioned to you was Goodstein's Theorem which Penrose has discussed in his article Can a Computer understand? - available in the book Mind, Matter and Mystery, Edited by Ranjit Nair.
Awesome...
Shencottah do you know of practical applications to this number?
I do not know, Phoenix.
It is quite interesting, Kovaiputhalvan.
Kovai,
Interesting m file....I notices that the convergence is not a increasing or decreasing function it oscillates in many cases ....whenever I think convergence I think a smooth curve (increasing or decreasing)
On Convergence of series
A series is (by notation) is Summation from n=1 to infinity of a_n. a_n are referred to as terms of the series. Let a_n be real numbers (series whose terms are not necessarily real numbers does make sense in normed linear spaces, see Limaye or Kreyszig's book on Functional analysis)
We know how to add finite number of numbers however large may be the finite number. So, when we see infinite sum anywhere, it needs to be given meaning. The meaning is given by saying that the above series represents that number to which the sequence s_N(called, sequence of partial sums) converges and s_N is defined as the sum of a_n with n varying from n=1 to N. That is, we 'cut the series' at n=N.
What Kovaiputhalavan said is correct, I restate the same: A real number s is said to be the limit of s_N as N goes to infinity if the terms s_N stay near s. Meaning, choose however small interval around s you want, the terms s_N belong to that interval after a certain stage N_0.
Of course, this N_0 itself depends on the interval given. This is what the English equivalent of Epsilon-Delta definition of limit.
Now, about the convergence of steps in the 6174 algorithm
At step 1, we start with a natural number A.
After step2, We still have a natural number(the number B). The numbers resulting after following Step 3 is also natural number. By repeating steps 1,2&3 we still end up with natural numbers.
Thus we have a sequence of natural numbers (sequence generated by a certain method (i.e., steps 1 to 4). We 'know' that it converges to 6174.
Now, recall the definition of convergence of a sequence. We woul like to see the nature of converging sequences of natural numbers. In other words, answer the question: "What are all the sequences of natural numbers that converge?" So, let s_n be a sequence of natural numbers. Suppose it converges to s (another number, possibly a real number). Take an interval centred at s of radius 1/4 (call it I). Since s_N converges to s,by definition of convergence, after certain stage N_0, the entire sequence belongs to the interval. Now the question is how many natural numbers are there in I? Note I is an interval of length 1/2. So, there can be atmost one natural number. What does this imply? This implies that s_N is equal to that natural number in I for n>=N_0.
That is s_n is eventually constant. It also means that s cannot be any real number but it has got to be a natural number (since I can take interval I containing s and such that I doesn't contain any natural number).
Finally...
Thus, we cannot really estimate the distance between 6174 and s_n generated by algorithm as s_n lives on natural numbers.
I dont know if there is a proof other than checking out the result 6174 for all the four digit numbers.
In one of the links given by Shencottah, there are generalisations of 6174 and in fact, in a paper (see the concluding remarks therein) http://www.cs.uwaterloo.ca/journals/JIS/VOL3/iann2a.html
the author tells what are all the Kaprekar numbers.
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