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Göbel's Sequence
In mathematics, a Göbel sequence is a sequence of rational numbers defined by the recurrence relation :x_n = \frac,\!\, with starting value :x_0 = x_1 = 1. Göbel's sequence starts with : 1, 1, 2, 3, 5, 10, 28, 154, 3520, 1551880, ... The first non-integral value is ''x''43. History This sequence was developed by the German mathematician Fritz Göbel in the 1970s. In 1975, the Dutch mathematician Hendrik Lenstra showed that the 43rd term is not an integer. Generalization Göbel's sequence can be generalized to ''k''th powers by :x_n = \frac. The least indices at which the ''k''-Göbel sequences assume a non-integral value are :43, 89, 97, 214, 19, 239, 37, 79, 83, 239, ... Regardless of the value chosen for ''k'', the initial 19 terms are always integers. See also * Somos sequence In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below. They were discovered by mathematician Michael Somos. From the form of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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 (for example, The set of all rational numbers is often referred to as "the rationals", and is closed under addition, subtraction, multiplication, and division by a nonzero rational number. It is a field under these operations and therefore also called the field of rationals or the field of rational numbers. It is usually denoted by boldface , or blackboard bold 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 n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recurrence Relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fritz Göbel
Fritz is a common German male name. The name originated as a German diminutive of Friedrich or Frederick (''Der Alte Fritz'', and ''Stary Fryc'' were common nicknames for King Frederick II of Prussia and Frederick III, German Emperor), as well as of similar names including Fridolin and, less commonly, Francis. Fritz (Fryc) was also a name given to German troops by Allies soldier similar to the term Tommy. Other common bases for which the name Fritz was used include the surnames Fritsche, Fritzsche, Fritsch, Frisch(e) and Frycz. Below is a list of notable people with the name "Fritz". Surname * Amanda Fritz (born 1958), retired registered psychiatric nurse and politician from Oregon *Al Fritz (1924–2013), American businessman * Ben Fritz (born 1981), American baseball coach * Betty Jane Fritz (1924–1994), one of the original players in the All-American Girls Professional Baseball League *Clemens Fritz (born 1980), German footballer * Edmund Fritz (before 1918–after 1932), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hendrik Lenstra
Hendrik Willem Lenstra Jr. (born 16 April 1949, Zaandam) is a Dutch mathematician. Biography Lenstra received his doctorate from the University of Amsterdam in 1977 and became a professor there in 1978. In 1987, he was appointed to the faculty of the University of California, Berkeley; starting in 1998, he divided his time between Berkeley and the University of Leiden, until 2003, when he retired from Berkeley to take a full-time position at Leiden. Three of his brothers, Arjen Lenstra, Andries Lenstra, and Jan Karel Lenstra, are also mathematicians. Jan Karel Lenstra is the former director of the Netherlands Centrum Wiskunde & Informatica (CWI). Hendrik Lenstra was the Chairman of the Program Committee of the International Congress of Mathematicians in 2010. Scientific contributions Lenstra has worked principally in computational number theory. He is well known for: * Co-discovering of the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (in 1982); * Developi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Somos Sequence
In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below. They were discovered by mathematician Michael Somos. From the form of their defining recurrence (which involves division), one would expect the terms of the sequence to be fractions, but surprisingly, a few Somos sequences have the property that all of their members are integers. Recurrence equations For an integer number ''k'' larger than 1, the Somos-''k'' sequence (a_0, a_1, a_2, \ldots ) is defined by the equation :a_n a_ = a_ a_ + a_ a_ + \cdots + a_ a_ when ''k'' is odd, or by the analogous equation :a_n a_ = a_ a_ + a_ a_ + \cdots + (a_)^2 when ''k'' is even, together with the initial values : ''a''''i'' = 1 for ''i'' < ''k''. For ''k'' = 2 or 3, these recursions are very simple (there is no addition on the right-hand side) and they define the all-ones sequence (1, 1, 1, 1, 1, 1, ...). In the first nontrivial case, ''k'' = 4, the defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Sequences
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description . The sequence 0, 3, 8, 15, ... is formed according to the formula ''n''2 − 1 for the ''n''th term: an explicit definition. Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number, , even though we do not have a formula for the ''n''th perfect number. Computable and definable sequences An integer sequence is computable if there exists an algorithm that, given '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
