|
Practical Number
In number theory, a practical number or panarithmic number is a positive integer n such that all smaller positive integers can be represented as sums of distinct divisors of n. For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, and 6: as well as these divisors themselves, we have 5 = 3 + 2, 7 = 6 + 1, 8 = 6 + 2, 9 = 6 + 3, 10 = 6 + 3 + 1, and 11 = 6 + 3 + 2. The sequence of practical numbers begins Practical numbers were used by Fibonacci in his Liber Abaci (1202) in connection with the problem of representing rational numbers as Egyptian fractions. Fibonacci does not formally define practical numbers, but he gives a table of Egyptian fraction expansions for fractions with practical denominators.. The name "practical number" is due to . He noted that "the subdivisions of money, weights, and measures involve numbers like 4, 12, 16, 20 and 28 which are usually supposed to be so inconvenient as to dese ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Practical Number Cuisenaire Rods 12
Pragmatism is a philosophical tradition that considers words and thought as tools and instruments for prediction, problem solving, and action (philosophy), action, and rejects the idea that the function of thought is to describe, represent, or mirror reality. Pragmatists contend that most philosophical topics—such as the nature of knowledge, language, concepts, meaning, belief, and science—are all best viewed in terms of their practical uses and successes. Pragmatism began in the United States in the 1870s. Its origins are often attributed to the philosophers Charles Sanders Peirce, William James, and John Dewey. In 1878, Peirce described it in his pragmatic maxim: "Consider the practical effects of the objects of your conception. Then, your conception of those effects is the whole of your conception of the object."Peirce, C.S. (1878), "s:How to Make Our Ideas Clear, How to Make Our Ideas Clear", ''Popular Science Monthly'', v. 12, 286–302. Reprinted often, including '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primorial
In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers. The name "primorial", coined by Harvey Dubner, draws an analogy to ''primes'' similar to the way the name "factorial" relates to ''factors''. Definition for prime numbers For the th prime number , the primorial is defined as the product of the first primes: :p_n\# = \prod_^n p_k, where is the th prime number. For instance, signifies the product of the first 5 primes: :p_5\# = 2 \times 3 \times 5 \times 7 \times 11 = 2310. The first five primorials are: : 2, 6, 30, 210, 2310 . The sequence also includes as empty product. Asymptotically, primorials grow according to: :p_n\# = e^, where is Little O notation. Definition for natural numbers In general, for a positive integer , its pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Prime
A Fibonacci prime is a Fibonacci number that is prime, a type of integer sequence prime. The first Fibonacci primes are : : 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, .... Known Fibonacci primes It is not known whether there are infinitely many Fibonacci primes. With the indexing starting with , the first 34 indices ''n'' for which ''F''''n'' is prime are : :''n'' = 3, 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, 2971, 4723, 5387, 9311, 9677, 14431, 25561, 30757, 35999, 37511, 50833, 81839, 104911, 130021, 148091. (Note that the actual values ''F''''n'' rapidly become very large, so, for practicality, only the indices are listed.) In addition to these proven Fibonacci primes, several probable primes have been found: :''n'' = 201107, 397379, 433781, 590041, 593689, 604711, 931517, 1049897, 1285607, 1636007, 1803059, 1968721, 2904353, 3244369, 3340367, 4740217, 6530879. 0 such that ''Fu'' is divisible by ''p'' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the first few values in the sequence are: :0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. The Fibonacci numbers were first described in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book ''Liber Abaci''. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the ''Fibonacci Quarterly''. Applications of Fibonacci numbers include co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giuseppe Melfi
Giuseppe Melfi (June 11, 1967) is an Italo-Swiss mathematician who works on practical numbers and modular forms. Career He gained his PhD in mathematics in 1997 at the University of Pisa. After some time spent at the University of Lausanne during 1997-2000, Melfi was appointed at the University of Neuchâtel, as well as at the University of Applied Sciences Western Switzerland and at the local University of Teacher Education. Work His major contributions are in the field of practical numbers. This prime-like sequence of numbers is known for having an asymptotic behavior and other distribution properties similar to the sequence of primes. Melfi proved two conjectures both raised in 1984 one of which is the corresponding of the Goldbach conjecture for practical numbers: every even number is a sum of two practical numbers. He also proved that there exist infinitely many triples of practical numbers of the form m-2,m,m+2. Another notable contribution has been in an application of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twin Prime Conjecture
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime'' is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger. However, it is unknown whether there are infinitely many twin primes (the so-called twin prime conjecture) or if there is a largest pair. The breakthrough work of Yitang Zhang in 2013, as well as work by James Maynard, Terence Tao and others, has made substantial progress towards proving that there are infinitely many twin primes, but at present this remains unsolved. Properties Usually the pair (2, 3) is not considered to be a pair of twin primes. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Goldbach's Conjecture
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to hold for all integers less than 4 × 1018, but remains unproven despite considerable effort. History On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII), in which he proposed the following conjecture: Goldbach was following the now-abandoned convention of considering 1 to be a prime number, so that a sum of units would indeed be a sum of primes. He then proposed a second conjecture in the margin of his letter, which implies the first: Euler replied in a letter dated 30 June 1742 and reminded Goldbach of an earlier conversation they had had (), in which Goldbach had remarked that the first of those two conjectures would follow from the statement This is in fact equivalent to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Numbers
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] |
Zhi-Wei Sun
Sun Zhiwei (, born October 16, 1965) is a Chinese mathematician, working primarily in number theory, combinatorics, and group theory. He is a professor at Nanjing University. Biography Sun Zhiwei was born in Huai'an, Jiangsu. Sun and his twin brother Sun Zhihong proved a theorem about what are now known as the Wall–Sun–Sun primes. Sun proved Sun's curious identity in 2002. In 2003, he presented a unified approach to three topics of Paul Erdős in combinatorial number theory: covering systems, restricted sumsets, and zero-sum problem In number theory, zero-sum problems are certain kinds of combinatorial problems about the structure of a finite abelian group. Concretely, given a finite abelian group ''G'' and a positive integer ''n'', one asks for the smallest value of ''k'' suc ...s or EGZ Theorem. With Stephen Redmond, he posed the Redmond–Sun conjecture in 2006. In 2013, he published a paper containing many conjectures on primes, one of which states that for any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Fraction
A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/''n''. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc. Arithmetic Elementary arithmetic Multiplying any two unit fractions results in a product that is another unit fraction: \frac1x \times \frac1y = \frac1. However, adding, subtracting, or dividing two unit fractions produces a result that is generally not a unit fraction: \frac1x + \frac1y = \frac \frac1x - \frac1y = \frac \frac1x \div \frac1y = \frac. Modular arithmetic In modular arithmetic, unit fractions can often be converted into equivalent integers using a calculation based on greatest common divisors. In turn, this conversion can be used to simplify division operations in modular arithmetic, by transforming them into equivalent multiplication operations. Specifically, consider the problem of dividing by a value x modulo y. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface , or blackboard bold \mathbb. A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real number that is not rational is called irrational. Irrational numbers include , , , and . Since the set of rational numbers is countable, and the set of real numbers is uncountable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \times (n-3) \times \cdots \times 3 \times 2 \times 1 \\ &= n\times(n-1)!\\ \end For example, 5! = 5\times 4! = 5 \times 4 \times 3 \times 2 \times 1 = 120. The value of 0! is 1, according to the convention for an empty product. Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book '' Sefer Yetzirah''. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, where its most basic use counts the possible distinct sequences – the permutations – of n distinct objects: there In mathematical analysis, factorials are used in power series for the exponential function an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
