Highly composite number
A composite number known as an anti-prime, is a positive integer with more divisors than any smaller positive integer has. The term was coined by Ramanujan. However, Jean-Pierre Kahane has suggested that the concept might have been known to Plato, who set 5040 as the ideal number of citizens in a city as 5040 has more divisors than any numbers less than it; the related concept of composite number refers to a positive integer which has at least as many divisors as any smaller positive integer. The name can be somewhat misleading, as two composite numbers are not composite numbers; the initial or smallest 38 composite numbers are listed in the table below. The number of divisors is given in the column labeled d; the table below shows all the divisors of one of these numbers. The 15,000th composite number can be found on Achim Flammenkamp's website, it is the product of 230 primes: a 0 14 a 1 9 a 2 6 a 3 4 a 4 4 a 5 3 a 6 3 a 7 3 a 8 2 a 9 2 a 10 2 a 11 2 a 12 2 a 13 2 a 14 2 a 15 2 a 16 2 a 17 2 a 18 2 a 19 a 20 a 21 ⋯ a 229, where a n is the sequence of successive prime numbers, all omitted terms are factors with exponent equal to one.
More concisely, it is the product of seven distinct primorials: b 0 5 b 1 3 b 2 2 b 4 b 7 b 18 b 229, where b n is the primorial a 0 a 1 ⋯ a n. Speaking, for a number to be composite it has to have prime factors as small as possible, but not too many of the same. By the fundamental theorem of arithmetic, every positive integer n has a unique prime factorization: n = p 1 c 1 × p 2 c 2 × ⋯ × p k c k where p 1 < p 2 < ⋯ < p k are prime, the exponents c i are positive integers. Any factor of n must have the same or lesser multiplicity in each prime: p 1 d 1 × p 2 d 2 × ⋯ × p k d k, 0 ≤ d i ≤ c i, 0 < i ≤ k So the number of divisors of n is: d = × × ⋯ ×. Hence, for a composite number n, the k given prime numbers pi must be the first k prime numbers. Except in two special cases n = 4 and n = 36, the last exponent ck must equal 1, it means that 1, 4, 36 are the only square com
A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1, not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 6 is composite because it is the product of two numbers that are both smaller than 6. 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, 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 n. Faster algorithms include the Miller–Rabin primality test, fast but has a small chance of error, the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical.
Fast methods are available for numbers of special forms, such as Mersenne numbers. As of December 2018 the largest known prime number has 24,862,048 decimal digits. There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled; the first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm. Several historical questions regarding prime numbers are still unsolved; these include Goldbach's conjecture, that every integer greater than 2 can be expressed as the sum of two primes, the twin prime conjecture, that there are infinitely many pairs of primes having just one number between them. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers.
Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals. A natural number is called a prime number if it is greater than 1 and cannot be written as a product of two natural numbers that are both smaller than it; the numbers greater than 1 that are not prime are called composite numbers. In other words, n is prime if n items cannot be divided up into smaller equal-size groups of more than one item, or if it is not possible to arrange n dots into a rectangular grid, more than one dot wide and more than one dot high. For example, among the numbers 1 through 6, the numbers 2, 3, 5 are the prime numbers, as there are no other numbers that divide them evenly. 1 is not prime, as it is excluded in the definition. 4 = 2 × 2 and 6 = 2 × 3 are both composite. The divisors of a natural number n are the numbers.
Every natural number has both itself as a divisor. If it has any other divisor, it cannot be prime; this idea leads to a different but equivalent definition of the primes: they are the numbers with two positive divisors, 1 and the number itself. Yet another way to express the same thing is that a number n is prime if it is greater than one and if none of the numbers 2, 3, …, n − 1 divides n evenly; the first 25 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. No number n greater than 2 is prime because any such number can be expressed as the product 2 × n / 2. Therefore, every prime number other than 2 is an odd number, is called an odd prime; when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are and decimal numbers that end in 0 or 5 are divisible by 5; the set of all primes is sometimes denoted by P or by P.
The Rhind Mathematical Papyrus, from around 1550 BC, has Egyptian fraction expansions of different forms for prime and composite numbers. However, the earliest surviving records of the explicit study of prime numbers come from Ancient Greek mathematics. Euclid's Elements proves the infinitude of primes and the fundamental theorem of arithmetic, shows how to construct a perfect number from a Mersenne prime. Another Greek invention, the Sieve of Eratosthenes, is still used to construct lists of primes. Around 1000 AD, the Islamic mathematician Alhazen found Wilson's theorem, characterizing the prime numbers as the numbers n that evenly divide
Gothenburg is the second-largest city in Sweden, fifth-largest in the Nordic countries, capital of the Västra Götaland County. It is situated by Kattegat, on the west coast of Sweden, has a population of 570,000 in the city center and about 1 million inhabitants in the metropolitan area. Gothenburg was founded as a fortified Dutch, trading colony, by royal charter in 1621 by King Gustavus Adolphus. In addition to the generous privileges given to his Dutch allies from the then-ongoing Thirty Years' War, the king attracted significant numbers of his German and Scottish allies to populate his only town on the western coast. At a key strategic location at the mouth of the Göta älv, where Scandinavia's largest drainage basin enters the sea, the Port of Gothenburg is now the largest port in the Nordic countries. Gothenburg is home to many students, as the city includes the University of Gothenburg and Chalmers University of Technology. Volvo was founded in Gothenburg in 1927; the original parent Volvo Group and the now separate Volvo Car Corporation are still headquartered on the island of Hisingen in the city.
Other key companies are Astra Zeneca. Gothenburg is served by Göteborg Landvetter Airport 30 km southeast of the city center; the smaller Göteborg City Airport, 15 km from the city center, was closed to regular airline traffic in 2015. The city hosts the Gothia Cup, the world's largest youth football tournament, alongside some of the largest annual events in Scandinavia; the Gothenburg Film Festival, held in January since 1979, is the leading Scandinavian film festival with over 155,000 visitors each year. In summer, a wide variety of music festivals are held in the city, including the popular Way Out West Festival; the city was named Göteborg in the city's charter in 1621 and given the German and English name Gothenburg. The Swedish name was given after the Göta älv, called Göta River in English, other cities ending in -borg. Both the Swedish and German/English names were in use before 1621 and had been used for the previous city founded in 1604 and burned down in 1611. Gothenburg is one of few Swedish cities to still have an official and used exonym.
Another example is the province of Scania in southern Sweden. The city council of 1641 consisted of four Swedish, three Dutch, three German, two Scottish members. In Dutch, Scots and German, all languages with a long history in this trade and maritime-oriented city, the name Gothenburg is or was used for the city. Variations of the official German/English name Gothenburg in the city's 1621 charter existed or exist in many languages; the French form of the city name is Gothembourg, but in French texts, the Swedish name Göteborg is more frequent. "Gothenburg" can be seen in some older English texts. In Spanish and Portuguese the city is called Gotemburgo; these traditional forms are sometimes replaced with the use of the Swedish Göteborg, for example by The Göteborg Opera and the Göteborg Ballet. However, Göteborgs universitet designated as the Göteborg University in English, changed its name to the University of Gothenburg in 2008; the Gothenburg municipality has reverted to the use of the English name in international contexts.
In 2009, the city council launched a new logotype for Gothenburg. Since the name "Göteborg" contains the Swedish letter "ö" the idea was to make the name more international and up to date by "turning" the "ö" sideways; as of 2015, the name is spelled "Go:teborg" on a large number of signs in the city. In the early modern period, the configuration of Sweden's borders made Gothenburg strategically critical as the only Swedish gateway to the North Sea and Atlantic, situated on the west coast in a narrow strip of Swedish territory between Danish Halland in the south and Norwegian Bohuslän in the north. After several failed attempts, Gothenburg was founded in 1621 by King Gustavus Adolphus; the site of the first church built in Gothenburg, subsequently destroyed by Danish invaders, is marked by a stone near the north end of the Älvsborg Bridge in the Färjenäs Park. The church was built in 1603 and destroyed in 1611; the city was influenced by the Dutch and Scots, Dutch planners and engineers were contracted to construct the city as they had the skills needed to drain and build in the marshy areas chosen for the city.
The town was designed like Dutch cities such as Amsterdam and New Amsterdam. The planning of the streets and canals of Gothenburg resembled that of Jakarta, built by the Dutch around the same time; the Dutchmen won political power, it was not until 1652, when the last Dutch politician in the city's council died, that Swedes acquired political power over Gothenburg. During the Dutch period, the town followed Dutch town laws and Dutch was proposed as the official language in the town. Robust city walls were built during the 17th century. In 1807, a decision was made to tear down most of the city's wall; the work started in 1810, was carried out by 150 soldiers from the Bohus regiment. Along with the Dutch, the town was influenced by Scots who settled down in Gothenburg. Many became people of high-profile. William Chalmers, the son of a Scottish immigrant, donated his fortunes to set up what became the Chalmers University of Technology. In 1841, the Scotsman Alexander Keiller founded the Götaverken shipbuilding company, in business until 1989.
His son James Keiller donated Keiller Park to the city in 1906. The Gothenburg coat of arms was based on the lion of the coat of arms of Sweden, symbolically holding a shield w
In number theory, a deficient or deficient number is a number n for which the sum of divisors σ<2n, or, the sum of proper divisors s<n. The value 2n − σ is called the number's deficiency; the first few deficient numbers are: 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50... As an example, consider the number 21, its proper divisors are 1, 3 and 7, their sum is 11. Because 11 is less than 21, the number 21 is deficient, its deficiency is 2 × 21 − 32 = 10. Since the aliquot sums of prime numbers equal 1, all prime numbers are deficient. An infinite number of both and odd deficient numbers exist. All odd numbers with one or two distinct prime factors are deficient. All proper divisors of deficient or perfect numbers are deficient. There exists at least one deficient number in the interval for all sufficiently large n. Related to deficient numbers are perfect numbers with σ = 2n, abundant numbers with σ > 2n.
The natural numbers were first classified as either deficient, perfect or abundant by Nicomachus in his Introductio Arithmetica. Perfect number Amicable number Sociable number Sándor, József. Handbook of number theory I. Dordrecht: Springer-Verlag. ISBN 1-4020-4215-9. Zbl 1151.11300. The Prime Glossary: Deficient number Weisstein, Eric W. "Deficient Number". MathWorld. Deficient number at PlanetMath.org
In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. The integer 12 is the first abundant number, its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance; the number 12 has an abundance of 4, for example. A number n for which the sum of divisors σ>2n, or, the sum of proper divisors s>n. Abundance is the value σ-2n; the first 28 abundant numbers are: 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, …. For example, the proper divisors of 24 are 1, 2, 3, 4, 6, 8, 12, whose sum is 36; because 36 is more than 24, the number 24 is abundant. Its abundance is 36 − 24 = 12; the smallest odd abundant number is 945. The smallest abundant number not divisible by 2 or by 3 is 5391411025 whose distinct prime factors are 5, 7, 11, 13, 17, 19, 23, 29. An algorithm given by Iannucci in 2005 shows how to find the smallest abundant number not divisible by the first k primes.
If A represents the smallest abundant number not divisible by the first k primes for all ϵ > 0 we have: 2 − ϵ < ln A < 2 + ϵ for sufficiently large k. Infinitely many and odd abundant numbers exist; the set of abundant numbers has a non-zero natural density. Marc Deléglise showed in 1998 that the natural density of the set of abundant numbers and perfect numbers is between 0.2474 and 0.2480. Every multiple of a perfect number is abundant. For example, every multiple of 6 is abundant because the divisors include 1, n/2, n/3, n/6 which sum to n + 1; every multiple of an abundant number is abundant. For example, every multiple of 20 is abundant because n/2 + n/4 + n/5 + n/10 + n/20 = n + n/10; every integer greater than 20161 can be written as the sum of two abundant numbers. An abundant number, not a semiperfect number is called a weird number. An abundant number with abundance 1 is called a quasiperfect number, although none have yet been found. Numbers whose sum of proper factors equals the number itself are called perfect numbers, while numbers whose sum of proper factors is less than the number itself are called deficient numbers.
The first known classification of numbers as deficient, perfect or abundant was by Nicomachus in his Introductio Arithmetica, which described abundant numbers as like deformed animals with too many limbs. The abundancy index of n is the ratio σ/n. Distinct numbers n1, n2... with the same abundancy index are called friendly numbers. The sequence of least numbers n such that σ > kn, in which a2 = 12 corresponds to the first abundant number, grows quickly. The smallest odd integer with abundancy index exceeding 3 is 1018976683725 = 33 × 52 × 72 × 11 × 13 × 17 × 19 × 23 × 29. If p = is a list of primes p is termed abundant if some integer composed only of primes in p is abundant. A necessary and sufficient condition for this is that the product of pi/ be at least 2. Tattersall, James J.. Elementary Number Theory in Nine Chapters. Cambridge University Press. ISBN 978-0-521-85014-8. Zbl 1071.11002. The Prime Glossary: Abundant number Weisstein, Eric W. "Abundant Number". MathWorld. Abundant number at PlanetMath.org
On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences cited as Sloane's, is an online database of integer sequences. It was maintained by Neil Sloane while a researcher at AT&T Labs. Foreseeing his retirement from AT&T Labs in 2012 and the need for an independent foundation, Sloane agreed to transfer the intellectual property and hosting of the OEIS to the OEIS Foundation in October 2009. Sloane is president of the OEIS Foundation. OEIS records information on integer sequences of interest to both professional mathematicians and amateurs, is cited; as of September 2018 it contains over 300,000 sequences. Each entry contains the leading terms of the sequence, mathematical motivations, literature links, more, including the option to generate a graph or play a musical representation of the sequence; the database is searchable by subsequence. Neil Sloane started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics; the database was at first stored on punched cards.
He published selections from the database in book form twice: A Handbook of Integer Sequences, containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372. The Encyclopedia of Integer Sequences with Simon Plouffe, containing 5,488 sequences and assigned M-numbers from M0000 to M5487; the Encyclopedia includes the references to the corresponding sequences in A Handbook of Integer Sequences as N-numbers from N0001 to N2372 The Encyclopedia includes the A-numbers that are used in the OEIS, whereas the Handbook did not. These books were well received and after the second publication, mathematicians supplied Sloane with a steady flow of new sequences; the collection became unmanageable in book form, when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service, soon after as a web site. As a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998; the database continues to grow at a rate of some 10,000 entries a year.
Sloane has managed'his' sequences for 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database. In 2004, Sloane celebrated the addition of the 100,000th sequence to the database, A100000, which counts the marks on the Ishango bone. In 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS.org was created to simplify the collaboration of the OEIS editors and contributors. The 200,000th sequence, A200000, was added to the database in November 2011. Besides integer sequences, the OEIS catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences. Sequences of rationals are represented by two sequences: the sequence of numerators and the sequence of denominators. For example, the fifth-order Farey sequence, 1 5, 1 4, 1 3, 2 5, 1 2, 3 5, 2 3, 3 4, 4 5, is catalogued as the numerator sequence 1, 1, 1, 2, 1, 3, 2, 3, 4 and the denominator sequence 5, 4, 3, 5, 2, 5, 3, 4, 5.
Important irrational numbers such as π = 3.1415926535897... are catalogued under representative integer sequences such as decimal expansions, binary expansions, or continued fraction expansions. The OEIS was limited to plain ASCII text until 2011, it still uses a linear form of conventional mathematical notation. Greek letters are represented by their full names, e.g. mu for μ, phi for φ. Every sequence is identified by the letter A followed by six digits always referred to with leading zeros, e.g. A000315 rather than A315. Individual terms of sequences are separated by commas. Digit groups are not separated by periods, or spaces. In comments, etc. A represents the nth term of the sequence. Zero is used to represent non-existent sequence elements. For example, A104157 enumerates the "smallest prime of n² consecutive primes to form an n×n magic square of least magic constant, or 0 if no such magic square exists." The value of a is 2. But there is no such 2×2 magic square, so a is 0; this special usage has a solid mathematical basis in certain counting functions.
For example, the totient valence function. There are 4 solutions for 4, but no solutions for 14, hence a of A014197 is 0—there are no solutions. −1 is used for this purpose instead, as in A094076. The OEIS ma
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to