|
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 numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 equal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of 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 are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle Detection
In computer science, cycle detection or cycle finding is the algorithmic problem of finding a cycle in a sequence of iterated function values. For any function that maps a finite set to itself, and any initial value in , the sequence of iterated function values : x_0,\ x_1=f(x_0),\ x_2=f(x_1),\ \dots,\ x_i=f(x_),\ \dots must eventually use the same value twice: there must be some pair of distinct indices and such that . Once this happens, the sequence must continue periodically, by repeating the same sequence of values from to . Cycle detection is the problem of finding and , given and . Several algorithms for finding cycles quickly and with little memory are known. Robert W. Floyd's tortoise and hare algorithm moves two pointers at different speeds through the sequence of values until they both point to equal values. Alternatively, Brent's algorithm is based on the idea of exponential search. Both Floyd's and Brent's algorithms use only a constant number of memory ce ... [...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 d ... [...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 0: digit_count = digit_count + 1 y = y // b total = 0 while x > 0: total = total + pow(x % b, digit_count) x = x // b return total def ppdif_cycle(x, b): seen = [] while x not in seen: seen.append(x) x = ppdif(x, b) cycle = [] while x not in cycle: cycle.append(x) x = ppdif(x, b) return cycle The following Python program determines whether the integer entered is a Narcissistic / Armstrong number or not. def no_of_digits(num): i = 0 while num > 0: num //= 10 i+=1 retu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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) = \sum_^ p_^. where k = \lfloor \log_ \rfloor + 1 is the number of digits in the number in base b, p_i is the i-prime number, 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 cycle of period k. A M ... [...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] |
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] |
Happy Number
In number theory, a happy number is a number which eventually reaches 1 when 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 digital invaria ... [...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 all ... [...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. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, -adic, and/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 or nonarchimedean dynamics, is an analogue of classical dynamics in which one replaces the complex numbers by a -adic field such as or and studies chaotic behavior and the Fatou and Julia sets. The following table describes a rough correspondence between Diophantine equations, espec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Python (programming Language)
Python is a high-level, general-purpose programming language. Its design philosophy emphasizes code readability with the use of significant indentation. Python is dynamically-typed and garbage-collected. It supports multiple programming paradigms, including structured (particularly procedural), object-oriented and functional programming. It is often described as a "batteries included" language due to its comprehensive standard library. Guido van Rossum began working on Python in the late 1980s as a successor to the ABC programming language and first released it in 1991 as Python 0.9.0. Python 2.0 was released in 2000 and introduced new features such as list comprehensions, cycle-detecting garbage collection, reference counting, and Unicode support. Python 3.0, released in 2008, was a major revision that is not completely backward-compatible with earlier versions. Python 2 was discontinued with version 2.7.18 in 2020. Python consistently ranks as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
