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Kaprekar
Dattatreya Ramchandra Kaprekar ( mr, दत्तात्रेय रामचंद्र कापरेकर; 17 January 1905 – 1986) was an Indian recreational mathematician who described several classes of natural numbers including the Kaprekar, harshad and self numbers and discovered the Kaprekar's constant, named after him. Despite having no formal postgraduate training and working as a schoolteacher, he published extensively and became well known in recreational mathematics circles. Biography Kaprekar received his secondary school education in Thane and studied at Cotton College in Guwahati. In 1927, he won the Wrangler R. P. Paranjpe Mathematical Prize for an original piece of work in mathematics. He attended the University of Mumbai, receiving his bachelor's degree in 1929. Having never received any formal postgraduate training, for his entire career (1930–1962) he was a schoolteacher at the government junior school in Devlali Maharashtra, India. Cyc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaprekar's Constant
In number theory, Kaprekar's routine is an iterative algorithm that, with each iteration, takes a natural number in a given number base, creates two new numbers by sorting the digits of its number by descending and ascending order, and subtracts the second from the first to yield the natural number for the next iteration. It is named after its inventor, the Indian mathematician D. R. Kaprekar. Kaprekar showed that in the case of four-digit numbers in base 10, if the initial number has at least two distinct digits, after seven iterations this process always yields the number 6174, which is now known as Kaprekar's constant. Definition and properties The algorithm is as follows: # Choose any natural number n in a given number base b. This is the first number of the sequence. # Create a new number \alpha by sorting the digits of n in descending order, and another new number \beta by sorting the digits of n in ascending order. These numbers may have leading zeros, which are discarded ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaprekar's Routine
In number theory, Kaprekar's routine is an iterative algorithm that, with each iteration, takes a natural number in a given number base, creates two new numbers by sorting the digits of its number by descending and ascending order, and subtracts the second from the first to yield the natural number for the next iteration. It is named after its inventor, the Indian mathematician D. R. Kaprekar. Kaprekar showed that in the case of four-digit numbers in base 10, if the initial number has at least two distinct digits, after seven iterations this process always yields the number 6174, which is now known as Kaprekar's constant. Definition and properties The algorithm is as follows: # Choose any natural number n in a given number base b. This is the first number of the sequence. # Create a new number \alpha by sorting the digits of n in descending order, and another new number \beta by sorting the digits of n in ascending order. These numbers may have leading zeros, which are discarded ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaprekar Number
In mathematics, a natural number in a given number base is a p-Kaprekar number if the representation of its square in that base can be split into two parts, where the second part has p digits, that add up to the original number. The numbers are named after D. R. Kaprekar. Definition and properties Let n be a natural number. We define the Kaprekar function for base b > 1 and power p > 0 F_ : \mathbb \rightarrow \mathbb to be the following: :F_(n) = \alpha + \beta, where \beta = n^2 \bmod b^p and :\alpha = \frac A natural number n is a p-Kaprekar number if it is a fixed point for F_, which occurs if F_(n) = n. 0 and 1 are trivial Kaprekar numbers for all b and p, all other Kaprekar numbers are nontrivial Kaprekar numbers. For example, in base 10, 45 is a 2-Kaprekar number, because : \beta = n^2 \bmod b^p = 45^2 \bmod 10^2 = 25 : \alpha = \frac = \frac = 20 : F_(45) = \alpha + \beta = 20 + 25 = 45 A natural number n is a sociable Kaprekar number if it is a periodic point for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Demlo Number
In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book ''Recreations in the Theory of Numbers''. A repunit prime is a repunit that is also a prime number. Primes that are repunits in base-2 are Mersenne primes. As of March 2022, the largest known prime number , the largest probable prime ''R''8177207 and the largest elliptic curve primality prime ''R''49081 are all repunits. Definition The base-''b'' repunits are defined as (this ''b'' can be either positive or negative) :R_n^\equiv 1 + b + b^2 + \cdots + b^ = \qquad\mbox, b, \ge2, n\ge1. Thus, the number ''R''''n''(''b'') consists of ''n'' copies of the digit 1 in base-''b'' representation. The first two repunits base-''b'' for ''n'' = 1 and ''n'' = 2 are :R_1^ 1 \qquad \text \qquad R_2^ b+1\qquad\text\ , b, \ge2. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Repunit
In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book ''Recreations in the Theory of Numbers''. A repunit prime is a repunit that is also a prime number. Primes that are repunits in base-2 are Mersenne primes. As of March 2022, the largest known prime number , the largest probable prime ''R''8177207 and the largest elliptic curve primality prime ''R''49081 are all repunits. Definition The base-''b'' repunits are defined as (this ''b'' can be either positive or negative) :R_n^\equiv 1 + b + b^2 + \cdots + b^ = \qquad\mbox, b, \ge2, n\ge1. Thus, the number ''R''''n''(''b'') consists of ''n'' copies of the digit 1 in base-''b'' representation. The first two repunits base-''b'' for ''n'' = 1 and ''n'' = 2 are :R_1^ 1 \qquad \text \qquad R_2^ b+1\qquad\text\ , b, \ge2. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recreational Mathematics
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited to being an endeavor for amateurs, many topics in this field require no knowledge of advanced mathematics. Recreational mathematics involves mathematical puzzles and games, often appealing to children and untrained adults, inspiring their further study of the subject. The Mathematical Association of America (MAA) includes recreational mathematics as one of its seventeen Special Interest Groups, commenting: Mathematical competitions (such as those sponsored by mathematical associations) are also categorized under recreational mathematics. Topics Some of the more well-known topics in recreational mathematics are Rubik's Cubes, magic squares, fractals, logic puzzles and mathematical chess problems, but this area of mathematics incl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recreational Mathematician
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited to being an endeavor for List of amateur mathematicians, amateurs, many topics in this field require no knowledge of advanced mathematics. Recreational mathematics involves mathematical puzzles and Mathematical games, games, often appealing to children and untrained adults, inspiring their further study of the subject. The Mathematical Association of America (MAA) includes recreational mathematics as one of its seventeen Mathematical Association of America#Special Interest Groups, Special Interest Groups, commenting: Mathematical competitions (such as those sponsored by Mathematical Association, mathematical associations) are also categorized under recreational mathematics. Topics Some of the more well-known topics in recreational ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
495 (number)
495 (four hundred ndninety-five) is the natural number following 494 and preceding 496. It is a pentatope number (and so a binomial coefficient \tbinom 4 ). The maximal number of pieces that can be obtained by cutting an annulus with 30 cuts. Kaprekar transformation The Kaprekar's routine algorithm is defined as follows for three-digit numbers: # Take any three-digit number, other than repdigits such as 111. Leading zeros are allowed. # Arrange the digits in descending and then in ascending order to get two three-digit numbers, adding leading zeros if necessary. # Subtract the smaller number from the bigger number. # Go back to step 2 and repeat. Repeating this process will always reach 495 in a few steps. Once 495 is reached, the process stops because 954 – 459 = 495. Example For example, choose 495: :495 The only three-digit numbers for which this function does not work are repdigits such as 111, which give the answer 0 after a single iteration. All other three- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self Number
In number theory, a self number or Devlali number in a given number base b is a natural number that cannot be written as the sum of any other natural number n and the individual digits of n. 20 is a self number (in base 10), because no such combination can be found (all n 1 F_b : \mathbb \rightarrow \mathbb to be the following: :F_(n) = n + \sum_^ d_i. where k = \lfloor \log_ \rfloor + 1 is the number of digits in the number in base b, and :d_i = \frac is the value of each digit of the number. A natural number n is a b-self number if the preimage of n for F_b is the empty set. In general, for even bases, all odd numbers below the base number are self numbers, since any number below such an odd number would have to also be a 1-digit number which when added to its digit would result in an even number. For odd bases, all odd numbers are self numbers.Sándor & Crstici (2004) p.384 The set of self numbers in a given base b is infinite and has a positive asymptotic density: when b i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harshad Number
In mathematics, a harshad number (or Niven number) in a given number base is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base are also known as -harshad (or -Niven) numbers. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "harshad" comes from the Sanskrit ' (joy) + ' (give), meaning joy-giver. The term "Niven number" arose from a paper delivered by Ivan M. Niven at a conference on number theory in 1977. Definition Stated mathematically, let be a positive integer with digits when written in base , and let the digits be a_i (i = 0, 1, \ldots, m-1). (It follows that a_i must be either zero or a positive integer up to .) can be expressed as :X=\sum_^ a_i n^i. is a harshad number in base if: :X \equiv 0 \bmod . A number which is a harshad number in every number base is called an all-harshad number, or an all-Niven number. There are only four all-harshad numbers: 1, 2, 4, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scientific American
''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it is the oldest continuously published magazine in the United States. ''Scientific American'' is owned by Springer Nature, which in turn is a subsidiary of Holtzbrinck Publishing Group. History ''Scientific American'' was founded by inventor and publisher Rufus Porter (painter), Rufus Porter in 1845 as a four-page weekly newspaper. The first issue of the large format newspaper was released August 28, 1845. Throughout its early years, much emphasis was placed on reports of what was going on at the United States Patent and Trademark Office, U.S. Patent Office. It also reported on a broad range of inventions including perpetual motion machines, an 1860 device for buoying vessels by Abraham Lincoln, and the universal joint which now can be found ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ivan M
Ivan () is a Slavic male given name, connected with the variant of the Greek name (English: John) from Hebrew meaning 'God is gracious'. It is associated worldwide with Slavic countries. The earliest person known to bear the name was Bulgarian tsar Ivan Vladislav. It is very popular in Russia, Ukraine, Croatia, Serbia, Bosnia and Herzegovina, Slovenia, Bulgaria, Belarus, North Macedonia, and Montenegro and has also become more popular in Romance-speaking countries since the 20th century. Etymology Ivan is the common Slavic Latin spelling, while Cyrillic spelling is two-fold: in Bulgarian, Russian, Macedonian, Serbian and Montenegrin it is Иван, while in Belarusian and Ukrainian it is Іван. The Old Church Slavonic (or Old Cyrillic) spelling is . It is the Slavic relative of the Latin name , corresponding to English ''John''. This Slavic version of the name originates from New Testament Greek (''Iōánnēs'') rather than from the Latin . The Greek name is in tur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
