Kaprekar number
Encyclopedia
In mathematics
, a Kaprekar number for a given base is a non-negative integer
, the representation of whose square in that base can be split into two parts that add up to the original number again. For instance, 45 is a Kaprekar number, because 45² = 2025 and 20+25 = 45. The Kaprekar numbers are named after D. R. Kaprekar
.
Note that X is also a Kaprekar no for base bn, for this specific choice of n. More narrowly, we can define the set K(N) for a given integer N as the set of integers X for which
Each Kaprekar number X for base b is then counted in one of the sets K(b), K(b²), K(b³),….
The first few Kaprekar numbers in base 10 are:
In particular, 9, 99, 999… are all Kaprekar numbers. More generally, for any base b, there exist infinitely many Kaprekar numbers, including all numbers of the form bn - 1.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Kaprekar number for a given base is a non-negative integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
, the representation of whose square in that base can be split into two parts that add up to the original number again. For instance, 45 is a Kaprekar number, because 45² = 2025 and 20+25 = 45. The Kaprekar numbers are named after D. R. Kaprekar
D. R. Kaprekar
Dattaraya Ramchandra Kaprekar was an Indian mathematician who discovered several results in number theory, including a class of numbers and a constant named after him...
.
Definition
Let X be a non-negative integer. X is a Kaprekar number for base b if there exist non-negative integers n, A, and positive number B satisfying:- X² = Abn + B, where 0 < B < bn
- X = A + B
Note that X is also a Kaprekar no for base bn, for this specific choice of n. More narrowly, we can define the set K(N) for a given integer N as the set of integers X for which
- X² = AN + B, where 0 < B < N
- X = A + B
Each Kaprekar number X for base b is then counted in one of the sets K(b), K(b²), K(b³),….
Examples
297 is a Kaprekar number for base 10, because 297² = 88209, which can be split into 88 and 209, and 88 + 209 = 297. By convention, the second part may start with the digit 0, but must be positive. For example, 999 is a Kaprekar number for base 10, because 999² = 998001, which can be split into 998 and 001, and 998 + 001 = 999. But 100 is not; although 100² = 10000 and 100 + 00 = 100, the second part here is not positive.The first few Kaprekar numbers in base 10 are:
- 1, 9, 4545 (number)45 is the natural number following 44 and followed by 46.- In mathematics :Forty-five is a triangular number, a hexagonal and 16-gonal number, a Kaprekar number, and a Harshad number....
, 5555 (number)55 is the natural number following 54 and preceding 56.-Albania:*Gazeta 55, a newspaper*Constitution law 55, a law during Communist Albania.-Mathematics:...
, 9999 (number)99 is the natural number following 98 and preceding 100.-Mathematics:99 is the ninth repdigit, a palindromic number and a Kaprekar number...
, 297, 703, 999999 (number)Nine hundred and ninety-nine is the natural number following nine hundred ninety-eight and preceding one thousand. 999 is a Harshad number and a Kaprekar number. It is the largest 3 digit whole number in base 10....
, 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777, 9999, 17344, 22222, 38962, 77778, 82656, 95121, 99999, 142857142857 (number)142857 is the six repeating digits of 1/7, 0., and is the best-known cyclic number in base 10. If it is multiplied by 2, 3, 4, 5, or 6, the answer will be a cyclic permutation of itself, and will correspond to the repeating digits of 2/7, 3/7, 4/7, 5/7, or 6/7, respectively.- Calculations :- 22/7...
, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170, ...
In particular, 9, 99, 999… are all Kaprekar numbers. More generally, for any base b, there exist infinitely many Kaprekar numbers, including all numbers of the form bn - 1.
Properties
- It was shown in 2000 that the Kaprekar numbers for base b are in bijection with the unitary divisors of bn − 1, in the following sense. Let Inv(a,b) denote the multiplicative inverse of a modulo b, namely the least positive integer m such that
. Then, a number X is in the set K(N) (defined above) if and only if X = d Inv(d, (N-1)/d) for some unitary divisor d of N-1. In particular,
- For each X in K(N), N - X is in K(N).
- In binaryBinary numeral systemThe binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
, all even perfect numberPerfect numberIn number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...
s are Kaprekar numbers.