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Taxicab Number
In mathematics, the ''n''th taxicab number, typically denoted Ta(''n'') or Taxicab(''n''), is defined as the smallest integer that can be expressed as a sum of two ''positive'' integer cubes in ''n'' distinct ways. The most famous taxicab number is 1729 = Ta(2) = 13 + 123 = 93 + 103, also known as the Hardy–Ramanujan number. The name is derived from a conversation involving mathematicians G. H. Hardy and Srinivasa Ramanujan. As told by Hardy: History and definition The pairs of summands of the Hardy–Ramanujan number Ta(2) = 1729 were first mentioned by Bernard Frénicle de Bessy, who published his observation in 1657. 1729 was made famous as the first taxicab number in the early 20th century by a story involving Srinivasa Ramanujan in claiming it to be the smallest for his particular example of two summands. In 1938, G. H. Hardy and E. M. Wright proved that such numbers exist for all positive integers ''n'', and their proof is easily converted into a program to generat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Srinivasa Ramanujan - OPC - 2
Venkateswara (, ), also known as Venkatachalapati, Venkata, Balaji and Srinivasa, is a Hindu deity, described as a form or avatar of the god Vishnu. He is the presiding deity of Venkateswara Temple, Tirupati. His consorts, Padmavati and Bhudevi, are avatars of the goddess Lakshmi, the consort of Vishnu. Etymology and other names Venkateswara literally means "Lord of Venkata". The word is a combination of the words ''Venkata'' (the name of a hill in Andhra Pradesh) and ''iśvara'' ("Lord"). According to the ''Brahmanda'' and '' Bhavishyottara'' Puranas, the word "Venkata" means "destroyer of sins", deriving from the Sanskrit words ''vem'' (sins) and ''kata'' (power of immunity). Venkateswara is known by many names such as Srinivasa (''in whom Lakshmi dwells''), Narayana (''The Primordial One''), Perumal (''the great lord''), Malayappa (''the lord of the Hill'') and Govinda (Protector of Cows). In Tamil, he is commonly called "Elumalayan", meaning Lord of Seven Hills. In Telu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol, +) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication, and Division (mathematics), division. The addition of two Natural number, whole numbers results in the total or ''summation, sum'' of those values combined. For example, the adjacent image shows two columns of apples, one with three apples and the other with two apples, totaling to five apples. This observation is expressed as , which is read as "three plus two Equality (mathematics), equals five". Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers, and complex numbers. Addition belongs to arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can also be performed on abstract objects such as Euclidean vector, vec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematical Proceedings Of The Cambridge Philosophical Society
''Mathematical Proceedings of the Cambridge Philosophical Society'' is a mathematical journal published by Cambridge University Press for the Cambridge Philosophical Society. It aims to publish original research papers from a wide range of pure and applied mathematics. The journal, titled ''Proceedings of the Cambridge Philosophical Society'' before 1975, has been published since 1843. Abstracting and indexing The journal is abstracted and indexed in *MathSciNet *Science Citation Index Expanded *Scopus *ZbMATH Open See also *Cambridge Philosophical Society The Cambridge Philosophical Society (CPS) is a scientific society at the University of Cambridge. It was founded in 1819. The name derives from the medieval use of the word philosophy to denote any research undertaken outside the fields of law ... External linksofficial website References Academic journals associated with learned and professional societies Cambridge University Press academic journals Mathematics e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Paul Vojta
Paul Alan Vojta (born September 30, 1957) is an American mathematician, known for his work in number theory on Diophantine geometry and Diophantine approximation. Contributions In formulating Vojta's conjecture, he pointed out the possible existence of parallels between the Nevanlinna theory of complex analysis, and diophantine analysis in the circle of ideas around the Mordell conjecture and abc conjecture. This suggested the importance of the ''integer solutions'' (affine space) aspect of diophantine equations. Vojta wrote the .dvi-previewer xdvi. He also wrote a vi clone. Education and career He was an undergraduate student at the University of Minnesota, where he became a Putnam Fellow in 1977, and a doctoral student at Harvard University (1983). He currently is a professor in the Department of Mathematics at the University of California, Berkeley. Awards and honors In 2012 he became a fellow of the American Mathematical Society The American Mathematical Society (AM ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Relatively Prime
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also ''is prime to'' or ''is coprime with'' . The numbers 8 and 9 are coprime, despite the fact that neither—considered individually—is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing When the integers and are coprime, the standard way of expressing this fact in mathematical notation is to indicate that their greatest common divisor is one, by the formula or . In their 1989 textbook '' Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed an alter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cubefree
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-free, but is not, because 18 is divisible by . The smallest positive square-free numbers are Square-free factorization Every positive integer n can be factored in a unique way as n=\prod_^k q_i^i, where the q_i different from one are square-free integers that are pairwise coprime. This is called the ''square-free factorization'' of . To construct the square-free factorization, let n=\prod_^h p_j^ be the prime factorization of n, where the p_j are distinct prime numbers. Then the factors of the square-free factorization are defined as q_i=\prod_p_j. An integer is square-free if and only if q_i=1 for all i > 1. An integer greater than one is the kth power of another integer if and only if k is a divisor of all i such that q_i\neq 1. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Generalized Taxicab Number
In number theory, the generalized taxicab number is the smallest number — if it exists — that can be expressed as the sum of numbers to the th positive power in different ways. For and , they coincide with taxicab number. \begin \mathrm(1, 2, 2) &= 4 = 1 + 3 = 2 + 2 \\ \mathrm(2, 2, 2) &= 50 = 1^2 + 7^2 = 5^2 + 5^2 \\ \mathrm(3, 2, 2) &= 1729 = 1^3 + 12^3 = 9^3 + 10^3 \end The latter example is 1729, as first noted by Ramanujan. Euler showed that \mathrm(4, 2, 2) = 635318657 = 59^4 + 158^4 = 133^4 + 134^4. However, is not known for any :No positive integer is known that can be written as the sum of two 5th powers in more than one way, and it is not known whether such a number exists. See also *Cabtaxi number In number theory, the -th cabtaxi number, typically denoted , is defined as the smallest positive integer that can be written as the sum of two ''positive or negative or 0'' cubes in ways. Such numbers exist for all , which follows from the analo ... Refer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cabtaxi Number
In number theory, the -th cabtaxi number, typically denoted , is defined as the smallest positive integer that can be written as the sum of two ''positive or negative or 0'' cubes in ways. Such numbers exist for all , which follows from the analogous result for taxicab numbers. Known cabtaxi numbers Only 10 cabtaxi numbers are known : \begin \mathrm(1) =& \ 1 \\ &= 1^3 + 0^3 \\ pt \mathrm(2) =& \ 91 \\ &= 3^3 + 4^3 \\ &= 6^3 - 5^3 \\ pt \mathrm(3) =& \ 728 \\ &= 6^3 + 8^3 \\ &= 9^3 - 1^3 \\ &= 12^3 - 10^3 \\ pt \mathrm(4) =& \ 2741256 \\ &= 108^3 + 114^3 \\ &= 140^3 - 14^3 \\ &= 168^3 - 126^3 \\ &= 207^3 - 183^3 \\ pt \mathrm(5) =& \ 6017193 \\ &= 166^3 + 113^3 \\ &= 180^3 + 57^3 \\ &= 185^3 - 68^3 \\ &= 209^3 - 146^3 \\ &= 246^3 - 207^3 \\ pt \mathrm(6) =& \ 1412774811 \\ &= 963^3 + 804^3 \\ &= 1134^3 - 357^3 \\ &= 1155^3 - 504^3 \\ &= 1246^3 - 805^3 \\ &= 2115^3 - 2004^3 \\ &= 4746^3 - 4725^3 \\ pt \mathrm(7 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
John Leech (mathematician)
John Leech (21 July 1926 in Weybridge, Surrey – 28 September 1992 in Scotland) was a British mathematician working in number theory, geometry and combinatorial group theory. He is best known for his discovery of the Leech lattice In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by Er ... in 1965. He also discovered Ta(3) in 1957. Leech was married to Jenifer Haselgrove, a British radio scientist. References External links MacTutor History of Mathematics biography 20th-century British mathematicians 1926 births 1992 deaths {{UK-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bernard Frénicle De Bessy
Bernard ('' Bernhard'') is a French and West Germanic masculine given name. It has West Germanic origin and is also a surname. The name is attested from at least the 9th century. West Germanic ''Bernhard'' is composed from the two elements ''bern'' "bear" and ''hard'' "brave, hardy". Its native Old English cognate was ''Beornheard'', which was replaced or merged with the French form ''Bernard'' that was brought to England after the Norman Conquest. The name ''Bernhard'' was notably popular among Old Frisian speakers. Its wider use was popularized due to Saint Bernhard of Clairvaux (canonized in 1174). In Ireland, the name was an anglicized form of Brian. Geographical distribution Bernard is the second most common surname in France. As of 2014, 42.2% of all known bearers of the surname ''Bernard'' were residents of France (frequency 1:392), 12.5% of the United States (1:7,203), 7.0% of Haiti (1:382), 6.6% of Tanzania (1:1,961), 4.8% of Canada (1:1,896), 3.6% of Nigeria (1:12,221 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
