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Perfect Totient Number
In number theory, a perfect totient number is an integer that is equal to the sum of its iterated totients. That is, we apply the totient function to a number ''n'', apply it again to the resulting totient, and so on, until the number 1 is reached, and add together the resulting sequence of numbers; if the sum equals ''n'', then ''n'' is a perfect totient number. For example, there are six positive integers less than 9 and relatively prime to it, so the totient of 9 is 6; there are two numbers less than 6 and relatively prime to it, so the totient of 6 is 2; and there is one number less than 2 and relatively prime to it, so the totient of 2 is 1; and , so 9 is a perfect totient number. The first few perfect totient numbers are : 3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, ... . In symbols, one writes :\varphi^i(n) = \begin \varphi(n), &\text i = 1 \\ \varphi(\varphi^(n)), &\text i \geq 2 \end for the iterated totient function ... [...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] |
255 (number)
255 (two hundred ndfifty-five) is the natural number following 254 and preceding 256. In mathematics Its factorization makes it a sphenic number. Since 255 = 28 – 1, it is a Mersenne number (though not a pernicious one), and the fourth such number not to be a prime number. It is a perfect totient number, the smallest such number to be neither a power of three nor thrice a prime. Since 255 is the product of the first three Fermat primes, the regular 255-gon is constructible. In base 10, it is a self number. 255 is a repdigit in base 2 (11111111), in base 4 (3333), and in base 16 (FF). \left\ = 255. In computing 255 is a special number in some tasks having to do with computing. This is the maximum value representable by an eight-digit binary number, and therefore the maximum representable by an unsigned 8-bit byte (the most common size of byte, also called an octet), the smallest common variable size used in high level programming languages (bit being smaller, but rarely ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Integer Sequences
The ''Journal of Integer Sequences'' is a peer-reviewed open-access academic journal in mathematics, specializing in research papers about integer sequences. It was founded in 1998 by Neil Sloane. Sloane had previously published two books on integer sequences, and in 1996 he founded the On-Line Encyclopedia of Integer Sequences (OEIS). Needing an outlet for research papers concerning the sequences he was collecting in the OEIS, he founded the journal. Since 2002 the journal has been hosted by the David R. Cheriton School of Computer Science at the University of Waterloo, with Waterloo professor Jeffrey Shallit as its editor-in-chief. There are no page charges for authors, and all papers are free to all readers. The journal publishes approximately 50–75 papers annually.. In most years from 1999 to 2014, SCImago Journal Rank has ranked the ''Journal of Integer Sequences'' as a third-quartile journal in discrete mathematics and combinatorics. It is indexed by ''Mathematical Revie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ''Disquisitiones Arithmeticae'', published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modulo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for ''n'' = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case ''n'' = ''k'', ''then'' it must also hold for the next case ''n'' = ''k'' + 1. These two steps establish that the statement holds for every natural number ''n''. The base case does not necessarily begin with ''n'' = 0, but often with ''n'' = 1, and possibly with any fixed natural number ''n'' = ''N'', establishing the truth of the statement for all natu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Of 3
In mathematics, a power of three is a number of the form where is an integer – that is, the result of exponentiation with number three as the base and integer as the exponent. Applications The powers of three give the place values in the ternary numeral system. In graph theory, powers of three appear in the Moon–Moser bound on the number of maximal independent sets of an -vertex graph, and in the time analysis of the Bron–Kerbosch algorithm for finding these sets. Several important strongly regular graphs also have a number of vertices that is a power of three, including the Brouwer–Haemers graph (81 vertices), Berlekamp–van Lint–Seidel graph (243 vertices), and Games graph (729 vertices). In enumerative combinatorics, there are signed subsets of a set of elements. In polyhedral combinatorics, the hypercube and all other Hanner polytopes have a number of faces (not counting the empty set as a face) that is a power of three. For example, a , or squa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divisible
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder. Definition An integer is divisible by a nonzero integer if there exists an integer such that n=km. This is written as :m\mid n. Other ways of saying the same thing are that divides , is a divisor of , is a factor of , and is a multiple of . If does not divide , then the notation is m\not\mid n. Usually, is required to be nonzero, but is allowed to be zero. With this convention, m \mid 0 for every nonzero integer . Some definitions omit the requirement that m be nonzero. General Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4; they are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiple (mathematics)
In mathematics, a multiple is the product of any quantity and an integer. In other words, for the quantities ''a'' and ''b'', it can be said that ''b'' is a multiple of ''a'' if ''b'' = ''na'' for some integer ''n'', which is called the multiplier. If ''a'' is not zero, this is equivalent to saying that b/a is an integer. When ''a'' and ''b'' are both integers, and ''b'' is a multiple of ''a'', then ''a'' is called a divisor of ''b''. One says also that ''a'' divides ''b''. If ''a'' and ''b'' are not integers, mathematicians prefer generally to use integer multiple instead of ''multiple'', for clarification. In fact, ''multiple'' is used for other kinds of product; for example, a polynomial ''p'' is a multiple of another polynomial ''q'' if there exists third polynomial ''r'' such that ''p'' = ''qr''. In some texts, "''a'' is a submultiple of ''b''" has the meaning of "''a'' being a unit fraction of ''b''" or, equivalently, "''b'' being an integer multiple of ''a''". This termino ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
729 (number)
700 (seven hundred) is the natural number following 699 and preceding 701. It is the sum of four consecutive primes (167 + 173 + 179 + 181), the perimeter of a Pythagorean triangle (75 + 308 + 317) and a Harshad number. Integers from 701 to 799 Nearly all of the palindromic integers between 700 and 800 are used as model numbers for Boeing Commercial Airplanes, and the only one not officially used by Boeing, 797, is commonly speculated to be the number of the next new Boeing commercial airplane. 700s * 701 = prime number, sum of three consecutive primes (229 + 233 + 239), Chen prime, Eisenstein prime with no imaginary part * 702 = 2 × 33 × 13, pronic number, nontotient, Harshad number * 703 = 19 × 37, triangular number, hexagonal number, smallest number requiring 73 fifth powers for Waring representation, Kaprekar number, area code for Northern Virginia along with 571, a number commonly found in the formula for body mass index * 704 = 26 × 11, Harshad number, lazy cater ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
363 (number)
363 (three hundred ndsixty-three) is the natural number following 362 and preceding 364. In mathematics * It is an odd, composite, positive, real integer, composed of a prime (3) and a prime squared (112). * 363 is a deficient number and a perfect totient number. * 363 is a palindromic number in bases 3, 10, 11 and 32. * 363 is a repdigit (BB) in base 32. * The Mertens function returns 0. * Any subset of its digits is divisible by three. * 363 is the sum of nine consecutive primes (23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59). * 363 is the sum of five consecutive powers of 3 (3 + 9 + 27 + 81 + 243). * 363 can be expressed as the sum of three squares in four different ways: 112 + 112 + 112, 52 + 72 + 172, 12 + 12 + 192, and 132 + 132 + 52. * 363 cubits is the solution given to Rhind Mathematical Papyrus question 50 – find the side length of an octagon with the same area as a circle 9 khet in diamete In other fields * The years AD 363 (in base 35 written AD AD) and 363 BC ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
327 (number)
300 (three hundred) is the natural number following 299 and preceding 301. Mathematical properties The number 300 is a triangular number and the sum of a pair of twin primes (149 + 151), as well as the sum of ten consecutive primes (13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47). It is palindromic in 3 consecutive bases: 30010 = 6067 = 4548 = 3639, and also in base 13. Factorization is 30064 + 1 is prime Other fields Three hundred is: * In bowling, a perfect score, achieved by rolling strikes in all ten frames (a total of twelve strikes) * The lowest possible Fair Isaac credit score * Three hundred ft/s is the maximum legal speed of a shot paintball * In the Hebrew Bible, the size of the military force deployed by the Israelite judge Gideon against the Midianites () * According to Islamic tradition, 300 is the number of ancient Israeli king Thalut's soldiers victorious against Goliath's soldiers * According to Herodotus, 300 is the number of ancient Spar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
