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17 January 2012

'गणितानंद' कापरेकर

'गणितानंद' कापरेकर

१७ जानेवारी १९०५ ला डहाणूत जन्मलेले द. रा. कापरेकर हे श्रीनिवास रामानुजन् यांच्यानंतरचे जागतिक कीर्तीचे गणितज्ज्ञ. १९५२ पर्यंत त्यांनी देवळालीत शिक्षकी केली..

नोकरी निमित्ताने कंटाळवाणा रेल्वेप्रवास करताना तिकिटावरच्या संख्येशी खेळता-खेळता ‘डेम्लो’ संख्येच्या रूपाने त्यांच्या हाती घबाड लागले.

‘मूषक उड्डाण उपपत्री’ आणि ‘कापरेकर स्थिरांक’, ‘जादूचे चौरस’, ‘संयोग संख्या’, ‘वानरी संख्या’, ‘हस्तमघद संख्या’ अशा त्यांनी शोधलेल्या कितीतरी संख्या आहेत. अनेक लेख, पुस्तके लिहिणाऱ्या कापरेकरांना जगभरात मान्यता मिळाली.

Kaparekar Number:
He showed that 6174 is reached in the limit as one repeatedly subtracts the highest and lowest numbers that can be constructed from a set of four digits that are not all identical. Thus, starting with 1234, we have
4321 - 1234 = 3087, then
8730 - 0378 = 8352, and
8532 - 2358 = 6174.
Repeating from this point onward leaves the same number (7641 - 1467 = 6174). In general, when the operation converges it does so in at most seven iterations.
A similar constant for 3 digits is 495. However, in base 10 a single such constant only exists for numbers of 3 or 4 digits; for more digits (or 2), the numbers enter into one of several cycles.

Kaprekar Constant:
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 number 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)

A Kaprekar number is a positive integer with the property that if it is squared, then its representation can be partitioned into two positive integer parts whose sum is equal to the original number e.g. 45, since 452=2025, and 20+25=45, also 9, 55, 99 etc

Devlali or Self number:

which are integers that cannot be generated by taking some other number and adding its own digits to it. For example, 21 is not a self number, since it can be generated from 15: 15 + 1 + 5 = 21. But 20 is a self number, since it cannot be generated from any other integer.

Harshad number:
These are defined by the property that they are divisible by the sum of their digits. Thus 12, which is divisible by 1 + 2 = 3, is a Harshad number.

Demlo number
Kaprekar also studied the Demlo numbers, named after a train station where he had the idea of studying them. These are the numbers 1, 121, 12321, …, which are the squares of the repunits 1, 11, 111.


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