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Perfect Digit-to-digit Invariant
In number theory, a perfect digit-to-digit invariant (PDDI; also known as a Munchausen number) is a natural number in a given number base b that is equal to the sum of its digits each raised to the power of itself. An example in base 10 is 3435, because 3435 = 3^3 + 4^4 + 3^3 + 5^5. The term "Munchausen number" was coined by Dutch mathematician and software engineer Daan van Berkel in 2009, as this evokes the story of Baron Munchausen raising himself up by his own ponytail because each digit is raised to the power of itself.Daan van Berkel,''On a curious property of 3435.''/ref> Definition Let n be a natural number which can be written in base b as the k-digit number d_d_...d_d_ where each digit d_i is between 0 and b-1 inclusive, and n = \sum_^ d_b^. We define the function F_b : \mathbb \rightarrow \mathbb as F_b(n) = \sum_^ ^. (As 00 is usually undefined, there are typically two conventions used, one where it is taken to be equal to one, and another where it is taken to be equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dudeney Number
In number theory, a Dudeney number in a given number base b is a natural number equal to the perfect cube of another natural number such that the digit sum of the first natural number is equal to the second. The name derives from Henry Dudeney, who noted the existence of these numbers in one of his puzzles, ''Root Extraction'', where a professor in retirement at Colney Hatch postulates this as a general method for root extraction. Mathematical definition Let n be a natural number. We define the Dudeney function for base b > 1 and power p > 0 F_ : \mathbb \rightarrow \mathbb to be the following: :F_(n) = \sum_^ \frac where k = p\left(\lfloor \log_ \rfloor + 1\right) is the p times the number of digits in the number in base b. A natural number n is a Dudeney root if it is a fixed point for F_, which occurs if F_(n) = n. The natural number m = n^p is a generalised Dudeney number, and for p = 3, the numbers are known as Dudeney numbers. 0 and 1 are trivial Dudeney numbers for al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brady Haran
Brady John Haran (born 18 June 1976) is an Australian-British independent filmmaker and video journalist who produces educational videos and documentary films for his YouTube channels, the most notable being ''Computerphile'' and ''Numberphile''. Haran is also the co-host of the'' Hello Internet'' podcast along with fellow educational YouTuber CGP Grey. On 22 August 2017, Haran launched his second podcast, called ''The Unmade Podcast'', and on 11 November 2018, he launched his third podcast, '' The Numberphile Podcast'', based on his mathematics-centered channel of the same name. Reporter and filmmaker Brady Haran studied journalism for a year before being hired by '' The Adelaide Advertiser''. In 2002, he moved from Australia to Nottingham, United Kingdom. In Nottingham, he worked for the BBC, began to work with film, and reported for '' East Midlands Today'', BBC News Online and BBC radio stations. In 2007, Haran worked as a filmmaker-in-residence for Nottingham Science C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sum-product Number
A sum-product number in a given number base b is a natural number that is equal to the product of the sum of its digits and the product of its digits. There are a finite number of sum-product numbers in any given base b. In base 10, there are exactly four numbers : 0, 1, 135, and 144. Definition Let n be a natural number. We define the sum-product function for base b > 1, F_b : \mathbb \rightarrow \mathbb, to be the following: : F_b(n) = \left(\sum_^k d_i\right)\!\!\left(\prod_^k d_j\right) where k = \lfloor \log_b \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 number if it is a fixed point for F_b, which occurs if F_b(n) = n. The natural numbers 0 and 1 are trivial numbers for all b, and all other numbers are nontrivial numbers. For example, the number 144 in base 10 is a sum-product number, because 1 + 4 + 4 = 9, 1 \times 4 \times 4 = 16, and 9 \times 16 = 144. A natural ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Digital Invariant
In number theory, a perfect digital invariant (PDI) is a number in a given number base (b) that is the sum of its own digits each raised to a given power (p). 0 F_ : \mathbb \rightarrow \mathbb is defined as: :F_(n) = \sum_^ d_i^p. 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 perfect digital invariant if it is a fixed point for F_, which occurs if F_(n) = n. 0 and 1 are trivial perfect digital invariants for all b and p, all other perfect digital invariants are nontrivial perfect digital invariants. For example, the number 4150 in base b = 10 is a perfect digital invariant with p = 5, because 4150 = 4^5 + 1^5 + 5^5 + 0^5. A natural number n is a sociable digital invariant if it is a periodic point for F_, where F_^k(n) = n for a positive integer k (here F_^k is the kth iterate of F_), and forms a cycle of period k. A perfect digital invariant is a sociable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Narcissistic Number
In number theory, a narcissistic number 1 F_ : \mathbb \rightarrow \mathbb to be the following: : F_(n) = \sum_^ d_i^k. 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 narcissistic number if it is a fixed point for F_, which occurs if F_(n) = n. The natural numbers 0 \leq n < b are trivial narcissistic numbers for all , all other narcissistic numbers are nontrivial narcissistic numbers. For example, the number 153 in base is a narcissistic number, because and . A natural number is a sociable narcissistic number if it is a for , where for a positive |
Meertens Number
In number theory and mathematical logic, a Meertens number in a given number base b is a natural number that is its own Gödel number. It was named after Lambert Meertens by Richard S. Bird as a present during the celebration of his 25 years at the CWI, Amsterdam. Definition Let n be a natural number. We define the Meertens function for base b > 1 F_ : \mathbb \rightarrow \mathbb to be the following: :F_(n) = \prod_^ p_^. where k = \lfloor \log_ \rfloor + 1 is the number of digits in the number in base b, p_i is the i-th prime number (starting at 0), and :d_i = \frac is the value of each digit of the number. A natural number n is a Meertens number if it is a fixed point for F_, which occurs if F_(n) = n. This corresponds to a Gödel encoding. For example, the number 3020 in base b = 4 is a Meertens number, because :3020 = 2^3^5^7^. A natural number n is a sociable Meertens number if it is a periodic point for F_, where F_^k(n) = n for a positive integer k, and forms a ... [...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. For example, in base 10, 45 is a 2-Kaprekar number, because 45² = 2025, and 20 + 25 = 45. The numbers are named after D. R. Kaprekar. Definition and properties Let n be a natural number. Then the Kaprekar function for base b > 1 and power p > 0 F_ : \mathbb \rightarrow \mathbb is defined 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. The earlier example of 45 satisfies this definition with b = 10 and p = 2, because : \beta = n^2 \bmod b^p = 45^2 \bmod 10^2 = 25 : \alpha = \frac = \frac = 20 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaprekar's Routine
In number theory, Kaprekar's routine is an iterative algorithm named after its inventor, Indian mathematician D. R. Kaprekar. Each iteration starts with a four-digit random number, sorts the digits into descending and ascending order, and calculates the difference between the two new numbers. As an example, starting with the number 8991 in base 10: : : : : 6174, known as Kaprekar's constant, is a fixed point of this algorithm. Any four-digit number (in base 10) with at least two distinct digits will reach 6174 within seven iterations. The algorithm runs on any natural number in any given number base. Definition and properties The algorithm is as follows: # Choose any four digit 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 number \beta by sorting the digits of n in ascending order. These numbers may have leading zeros, which can be ignored. Subtract ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Happy Number
In number theory, a happy number is a number which eventually reaches 1 when the number is replaced by the sum of the square of each digit. For instance, 13 is a happy number because 1^2+3^2=10, and 1^2+0^2=1. On the other hand, 4 is not a happy number because the sequence starting with 4^2=16 and 1^2+6^2=37 eventually reaches 2^2+0^2=4, the number that started the sequence, and so the process continues in an infinite cycle without ever reaching 1. A number which is not happy is called sad or unhappy. More generally, a b-happy number is a natural number in a given number base b that eventually reaches 1 when iterated over the perfect digital invariant function for p = 2. The origin of happy numbers is not clear. Happy numbers were brought to the attention of Reg Allenby (a British author and senior lecturer in pure mathematics at Leeds University) by his daughter, who had learned of them at school. However, they "may have originated in Russia" . Happy numbers and perfect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorion
In number theory, a factorion in a given number base b is a natural number that equals the sum of the factorials of its digits. The name factorion was coined by the author Clifford A. Pickover. Definition Let n be a natural number. For a base b > 1, we define the sum of the factorials of the digits of n, \operatorname_b : \mathbb \rightarrow \mathbb, to be the following: :\operatorname_b(n) = \sum_^ d_i!. where k = \lfloor \log_b n \rfloor + 1 is the number of digits in the number in base b, n! is the factorial of n and :d_i = \frac is the value of the ith digit of the number. A natural number n is a b-factorion if it is a fixed point for \operatorname_b, i.e. if \operatorname_b(n) = n. 1 and 2 are fixed points for all bases b, and thus are trivial factorions for all b, and all other factorions are nontrivial factorions. For example, the number 145 in base b = 10 is a factorion because 145 = 1! + 4! + 5!. For b = 2, the sum of the factorials of the digits is simply the num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Dynamics
Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Part of the inspiration comes from complex dynamics, the study of the Iterated function, iteration of self-maps of the complex plane or other complex algebraic varieties. Arithmetic dynamics is the study of the number-theoretic properties of integer point, integer, rational point, rational, p-adic number, -adic, or algebraic points under repeated application of a polynomial or rational function. A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures. ''Global arithmetic dynamics'' is the study of analogues of classical diophantine geometry in the setting of discrete dynamical systems, while ''local arithmetic dynamics'', also called p-adic dynamics, p-adic or nonarchimedean dynamics, is an analogue of complex dynamics in which one replaces the complex numbers by a -adic field such as or and studies chaotic behavior and the Fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
