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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] |
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] |
Base-2
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically "0" (zero) and "1" (one). A ''binary number'' may also refer to a rational number that has a finite representation in the binary numeral system, that is, the quotient of an integer by a power of two. The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implementation. History The modern binary number system was studied in Europe in the 16th and 17th centuries by Thomas Harriot, an ... [...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] |
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] |
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] |
Base-16
Hexadecimal (also known as base-16 or simply hex) is a positional numeral system that represents numbers using a radix (base) of sixteen. Unlike the decimal system representing numbers using ten symbols, hexadecimal uses sixteen distinct symbols, most often the symbols "0"–"9" to represent values 0 to 9 and "A"–"F" to represent values from ten to fifteen. Software developers and system designers widely use hexadecimal numbers because they provide a convenient representation of binary-coded values. Each hexadecimal digit represents four bits (binary digits), also known as a nibble (or nybble). For example, an 8-bit byte is two hexadecimal digits and its value can be written as to in hexadecimal. In mathematics, a subscript is typically used to specify the base. For example, the decimal value would be expressed in hexadecimal as . In programming, several notations denote hexadecimal numbers, usually involving a prefix. The prefix 0x is used in C, which would denote this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base-12
The duodecimal system, also known as base twelve or dozenal, is a positional numeral system using twelve as its base. In duodecimal, the number twelve is denoted "10", meaning 1 twelve and 0 units; in the decimal system, this number is instead written as "12" meaning 1 ten and 2 units, and the string "10" means ten. In duodecimal, "100" means twelve squared (144), "1,000" means twelve cubed (1,728), and "0.1" means a twelfth (0.08333...). Various symbols have been used to stand for ten and eleven in duodecimal notation; this page uses and , as in hexadecimal, which make a duodecimal count from zero to twelve read 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, , , and finally 10. The Dozenal Societies of America and Great Britain (organisations promoting the use of duodecimal) use turned digits in their published material: (a turned 2) for ten (dek, pronounced dɛk) and (a turned 3) for eleven (el, pronounced ɛl). The number twelve, a superior highly composite number, is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base-10
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as ''decimal notation''. A decimal numeral (also often just ''decimal'' or, less correctly, ''decimal number''), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by a decimal separator (usually "." or "," as in or ). ''Decimal'' may also refer specifically to the digits after the decimal separator, such as in " is the approximation of to ''two decimals''". Zero-digits after a decimal separator serve the purpose of signifying the precision of a value. The numbers that may be represented in the decimal system are the decimal fractions. That is, fractions of the form , where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base-9
A ternary numeral system (also called base 3 or trinary) has three as its base. Analogous to a bit, a ternary digit is a trit (trinary digit). One trit is equivalent to log2 3 (about 1.58496) bits of information. Although ''ternary'' most often refers to a system in which the three digits are all non–negative numbers; specifically , , and , the adjective also lends its name to the balanced ternary system; comprising the digits −1, 0 and +1, used in comparison logic and ternary computers. Comparison to other bases Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary. For example, decimal 365 or senary corresponds to binary (nine bits) and to ternary (six digits). However, they are still far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary (base 9) and septemvigesimal (base 27). : : : As for rational numbers, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base-8
Octal (base 8) is a numeral system with eight as the base. In the decimal system, each place is a power of ten. For example: : \mathbf_ = \mathbf \times 10^1 + \mathbf \times 10^0 In the octal system, each place is a power of eight. For example: : \mathbf_8 = \mathbf \times 8^2 + \mathbf \times 8^1 + \mathbf \times 8^0 By performing the calculation above in the familiar decimal system, we see why 112 in octal is equal to 64+8+2=74 in decimal. Octal numerals can be easily converted from binary representations (similar to a quaternary numeral system) by grouping consecutive binary digits into groups of three (starting from the right, for integers). For example, the binary representation for decimal 74 is 1001010. Two zeroes can be added at the left: , corresponding to the octal digits , yielding the octal representation 112. Usage In China The eight bagua or trigrams of the I Ching correspond to octal digits: * 0 = ☷, 1 = ☳, 2 = ☵, 3 = ☱, * 4 = ☶, 5 = ☠... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base-6
A senary () numeral system (also known as base-6, heximal, or seximal) has six as its base. It has been adopted independently by a small number of cultures. Like the decimal base 10, the base is a semiprime, though it is unique as the product of the only two consecutive numbers that are both prime (2 and 3). As six is a superior highly composite number, many of the arguments made in favor of the duodecimal system also apply to the senary system. Formal definition The standard set of digits in the senary system is \mathcal_6 = \lbrace 0, 1, 2, 3, 4, 5\rbrace, with the linear order 0 < 1 < 2 < 3 < 4 < 5. Let be the of , where is the operation of |
