The Kaprekar Numbers

Douglas E. Iannucci
The University of the Virgin Islands
2 John Brewers Bay
St. Thomas, VI 00802
Email address: diannuc@uvi.edu

Abstract: The (decimal) n-Kaprekar numbers are defined and are shown to be in one-one correspondence with the unitary divisors of 10ⁿ - 1. In particular, this establishes the correctness of an algorithm for generating the Kaprekar numbers proposed by Charosh in 1981. The even perfect numbers are shown to be Kaprekar numbers in the binary base.

1. INTRODUCTION

The Kaprekar numbers (sequence A006886 in [4]) were introduced by the eponymous D. R. Kaprekar [3] in 1980. They have been the subject of several articles, and are mentioned in David Wells's Dictionary of Curious and Interesting Numbers [5].
What makes the Kaprekar numbers curious or interesting? Let's consider an example. The number 703 is Kaprekar because

703² = 494209 and 494 + 209 = 703 .

Here are some further examples:

9² = 81 , 8 + 1 = 9 ;
45² = 2025 , 20 + 25 = 45 ;
297² = 88209 , 88 + 209 = 297 ;
4879² = 23804641 , 238 +たす 4641 =わ 4879 ;
17344² = 300814336 , 3008 +たす 14336 =わ 17344 ;
538461² = 289940248521 , 289940 +たす 248521 =わ 538461 .

Formally, an n-Kaprekar number k >= 1 (for n = 1, 2, ...) satisfies the pair of equations

k = q + r ,

k² = q * 10ⁿ + r

where q >= 1 and 0 <= r < 10ⁿ. As the 5-Kaprekar number k = 4879 shows, r may have fewer than n digits. We adopt the convention that 1 is an n-Kaprekar number for all n >= 1, since

1² = 0 * 10ⁿ + 1, 1 = 0 + 1 ;

but by fiat 0 and 10^m for m >= 1 are not Kaprekar numbers.
Kaprekar [3] listed 9 as a Kaprekar number, but failed to list 99, 999, ... However, 10ⁿ - 1 (for all n >= 1) is n-Kaprekar since

99 . . . 99² = 9 . . . 9980 . . . 001 , 9 . . . 998 + 0 . . . 001 = 99 . . . 99 ;
\______/ \______/ \_____/ \_____/ \_____/ \______/
n nines n-1 nines n-1 zeros n-1 nines n-1 zeros n nines

that is,

(10ⁿ - 1)² = (10ⁿ - 2) * 10ⁿ + 1 , 10ⁿ - 1 = (10ⁿ - 2) + 1 .

Charosh [2] noted Kaprekar's omission of the numbers 10ⁿ - 1 (n >= 1), as well as the 6-Kaprekar numbers 181819 and 818181. In fact, Charosh correctly devised a method by which to construct Kaprekar numbers of any size. In this paper, we will refine Charosh's result by establishing a bijection between the n-Kaprekar numbers and the unitary divisors of 10ⁿ -1 (thus refining and proving Charosh's result). Recall that a is a unitary divisor of m if ab = m and (a, b) = 1 .

2. THE MAIN RESULT

For each integer N > 1, let K(N) denote the set of positive integers k for which there exists integers q and r such that

As a matter of convention, we shall ignore the vacuous solution k = N (for which q = N and r = 0). (1) and (2) imply

Since we disregard the vacuous solution, we have 1 <= k <= N - 1 (for if k >= N then (3) implies q > k, contradicting (2)).
The set K(N) is nonempty, for always 1 is in K(N). Suppose k were in K(N). Since (k, k - 1) = 1, it follows from (3) that d | k and d' | k - 1 for some positive d and d' such that dd' = N - 1 and (d, d' ) = 1. Let k' = N - k . Because 1 <= k <= N - 1 , we have k' > 0 . Since k' = (N - 1) - (k - 1) , we have d' | k' . Thus k = dm and k' = d'm' for some positive m and m' , whence follows

Applying Lemma 1 to (4) gives

Conversely, let dd' = N - 1 , (d, d') = 1, and let m = Inv(d, d' ) and m' = Inv(d', d ) . Then by Lemma 1 we have dm + d'm' = N . Therefore

d²m² = (N - d'm')²
= N ² - N d'm' - (dm + d'm') d'm' + (d'm')²
= N ² - N d'm' - mm'dd'
= (N - d'm' - mm') N + mm' .

Thus
(dm)² = (N - d'm' - mm') N + mm' (mm' < N) ,
dm = (N - d'm' - mm') + mm' .

That is, dm satisfies (1) and (2) (with q = N - d'm' - mm' and r = mm'), whence dm is in K(N). Note that d'm' is in K(N) by symmetry. We have proved the following results:

Let w(M) denote the number of distinct primes dividing M; then M has exactly 2^w(M) unitary divisors. The following result is immediate:

The convention that N not be an element of K(N) was taken to ensure the bijection between the elements of K(N) and the unitary divisors of N - 1 .

3. APPLICATIONS

If we let N = 10ⁿ for some n >= 1 in Theorem 1, we get the set of n-Kaprekar numbers, which is thus given by

K(10ⁿ) = { d Inv(d , d' ) : dd' = 10ⁿ - 1 , (d , d' ) = 1 }.

The following table lists all n-Kaprekar numbers k for 1 <= n <= 6, along with the associated unitary divisors d of 10ⁿ - 1 . These same Kaprekar numbers were given by Charosh [2]. By Corollary A, the n-Kaprekar numbers occur in complementary pairs which sum to 10ⁿ .

For example, consider n = 3: 27 and 37 are unitary divisors of 10³ - 1 = 27 * 37. Then Inv(27, 37) = 11 and Inv(37, 27) = 19, and we obtain the complementary 3-Kaprekar numbers 27 * 11 = 297 and 37 * 19 = 703.
The universal Kaprekar number 1 corresponds to the unitary divisor 1 of 10ⁿ - 1, which is why we allow unity as a Kaprekar number. For each n >= 1 , we disallow 10ⁿ as a Kaprekar number since it is the vacuous solution to (1) and (2) when N = 10ⁿ .
If we let N = bⁿ in Theorem 1 for some b >= 2 and n >= 1 , we get the base-b generalization of the Kaprekar numbers. The case b = 2 is especially interesting.

Proof: Let n >= 1. It is clear that 2ⁿ - 1 and 2ⁿ + 1 are relatively prime, and that

2^n-1 (2ⁿ - 1) = 1 (mod 2ⁿ + 1) , 0 < 2^n-1 < 2ⁿ + 1 .

Therefore 2^n-1 = Inv( 2ⁿ - 1 , 2ⁿ + 1), and so 2^n-1 (2ⁿ - 1) is in K(2²ⁿ)by Theorem 1. It is well known that every even perfect number has the form 2^n-1 (2ⁿ - 1) where 2ⁿ - 1 is prime; hence the result follows. QED

To illustrate Theorem 2, we see that 28 = 2² (2³ - 1) is perfect and 6-Kaprekar in the binary base: (28)₂ = 11100, and

11100² = 1100010000 , 1100 +たす 010000 =わ 11100 .

Similarly, b^n-1 (bⁿ - 1) and b^n-1 (bⁿ + 1) are complementary 2n-Kaprekar numbers in the base b whenever b is even. This pattern, among others, was noted by Charosh [2] in the case when b = 10.

4. CONCLUDING REMARKS
It is not worth compiling a more extensive list of Kaprekar numbers, since they can be obtained from the prime factorization of 10ⁿ - 1 cf. Brillhart et al. [1].
Corollary B shows that the Kaprekar numbers have natural density zero.

REFERENCES.

[1] Brillhart, J., Lehmer, D.H., Selfridge, J., Tuckerman, B., Wagstaff, S. "Factorizations of (bⁿ +/- 1) , b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers." 2nd ed., Contemporary Mathematics, v. 22, American Mathematical Society, Providence, RI, 1988.

[2] Charosh, M. "Some Applications of Casting Out 999...'s." Journal of Recreational Mathematics, v. 14, no. 2 (1981-82), pp. 111-118.

[3] Kaprekar, D. "On Kaprekar Numbers." Journal of Recreational Mathematics, v. 13, no. 2 (1980-81), pp. 81-82.

[4] Sloane, N.J.A. The On-Line Encyclopedia of Integer Sequences, published electronically at http://oeis.org

[5] Wells, D. The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books USA, Inc., New York 1986.

(Concerned with sequences A006886, A037042, A053394, A053395, A053396, A053397 .)

Received Feb 3, 1999; revised version received Mar 21, 1999. Published in Journal of Integer Sequences, Jan. 13, 2000.