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Taxicab Number
In mathematics, the ''n''th taxicab number, typically denoted Ta(''n'') or Taxicab(''n''), also called the ''n''th Hardy–Ramanujan number, 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. The name is derived from a conversation in about 1919 involving mathematicians G. H. Hardy and Srinivasa Ramanujan. As told by Hardy: History and definition The concept was first mentioned in 1657 by Bernard Frénicle de Bessy, who published the Hardy–Ramanujan number Ta(2) = 1729. This particular example of 1729 was made famous in the early 20th century by a story involving Srinivasa Ramanujan. 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 generate such numbers. However, the proof makes no claims at all about whether the thus-gen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Srinivasa Ramanujan - OPC - 2
Venkateswara, also known by various other names, is a form of the Hindu god Vishnu. Venkateswara is the presiding deity of the Tirumala Venkateswara Temple, located in Tirupati, Sri Balaji District, Andhra Pradesh, India. Etymology Venkateswara literally means, "Lord of Venkata". The word is a combination of the words ''Venkata'' (the name of a hill in Andhra Pradesh) and ''isvara'' ("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). It is also said that 'Venkata' is a combination of two words: '''ven''' (keeps away) and kata''' (troubles). Venkata means he 'who keeps away troubles' or 'who takes away problems' or such terms in a similar context. Legend Every year, hundreds of thousands of devotees donate a large amount of wealth at the Tirumala Venkateswara Temple at Tirupati, Andhra Pradesh. A legend provides the reason f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sums Of Powers
In mathematics and statistics, sums of powers occur in a number of contexts: * Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the analysis of variance involves summing the squares of quantities. *Faulhaber's formula expresses 1^k + 2^k + 3^k + \cdots + n^k as a polynomial in ''n'', or alternatively in term of a Bernoulli polynomial. *Fermat's right triangle theorem states that there is no solution in positive integers for a^2=b^4+c^4 and a^4=b^4+c^2. *Fermat's Last Theorem states that x^k+y^k=z^k is impossible in positive integers with ''k''>2. *The equation of a superellipse is , x/a, ^k+, y/b, ^k=1. The squircle is the case k=4, a=b. *Euler's sum of powers conjecture (disproved) concerns situations in which the sum of ''n'' integers, each a ''k''th power of an integer, equals another ''k''th p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sums Of Three Cubes
In the mathematics of sums of powers, it is an open problem to characterize the numbers that can be expressed as a sum of three cubes of integers, allowing both positive and negative cubes in the sum. A necessary condition for n to equal such a sum is that n cannot equal 4 or 5 modulo 9, because the cubes modulo 9 are 0, 1, and −1, and no three of these numbers can sum to 4 or 5 modulo 9. It is unknown whether this necessary condition is sufficient. Variations of the problem include sums of non-negative cubes and sums of rational cubes. All integers have a representation as a sum of rational cubes, but it is unknown whether the sums of non-negative cubes form a set with non-zero natural density. Small cases A nontrivial representation of 0 as a sum of three cubes would give a counterexample to Fermat's Last Theorem for the exponent three, as one of the three cubes would have the opposite sign as the other two and its negation would equal the sum of the other two. Therefo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagorean Quadruple
A Pythagorean quadruple is a tuple of integers , , , and , such that . They are solutions of a Diophantine equation and often only positive integer values are considered.R. Spira, ''The diophantine equation '', Amer. Math. Monthly Vol. 69 (1962), No. 5, 360–365. However, to provide a more complete geometric interpretation, the integer values can be allowed to be negative and zero (thus allowing Pythagorean triples to be included) with the only condition being that . In this setting, a Pythagorean quadruple defines a cuboid with integer side lengths , , and , whose space diagonal has integer length ; with this interpretation, Pythagorean quadruples are thus also called ''Pythagorean boxes''. In this article we will assume, unless otherwise stated, that the values of a Pythagorean quadruple are all positive integers. Parametrization of primitive quadruples A Pythagorean quadruple is called primitive if the greatest common divisor of its entries is 1. Every Pythagorean quadrup ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prouhet–Tarry–Escott Problem
In mathematics, the Prouhet–Tarry–Escott problem asks for two disjoint multisets ''A'' and ''B'' of ''n'' integers each, whose first ''k'' power sum symmetric polynomials are all equal. That is, the two multisets should satisfy the equations :\sum_ a^i = \sum_ b^i for each integer ''i'' from 1 to a given ''k''. It has been shown that ''n'' must be strictly greater than ''k''. Solutions with k = n - 1 are called ''ideal solutions''. Ideal solutions are known for 3 \le n \le 10 and for n = 12. No ideal solution is known for n=11 or for n \ge 13. This problem was named after Eugène Prouhet, who studied it in the early 1850s, and Gaston Tarry and Edward B. Escott, who studied it in the early 1910s. The problem originates from letters of Christian Goldbach and Leonhard Euler (1750/1751). Examples Ideal solutions An ideal solution for ''n'' = 6 is given by the two sets and , because: : 01 + 51 + 61 + 161 + 171 + 221 = 11 + 21 + 101 + 121 + 201 + 211 : 02 + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi–Madden Equation
The Jacobi–Madden equation is the Diophantine equation : a^4 + b^4 + c^4 + d^4 = (a + b + c + d)^4 , proposed by the physicist Lee W. Jacobi and the mathematician Daniel J. Madden in 2008. The variables ''a'', ''b'', ''c'', and ''d'' can be any integers, positive, negative or 0. Jacobi and Madden showed that there are an infinitude of solutions of this equation with all variables non-zero. History The Jacobi–Madden equation represents a particular case of the equation : a^4 + b^4 +c^4 +d^4 = e^4 , first proposed in 1772 by Leonhard Euler who conjectured that four is the minimum number (greater than one) of fourth powers of non-zero integers that can sum up to another fourth power. This conjecture, now known as Euler's sum of powers conjecture, was a natural generalization of the Fermat's Last Theorem, the latter having been proved for the fourth power by Pierre de Fermat himself. Noam Elkies was first to find an infinite series of solutions to Euler's equation with ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beal's Conjecture
The Beal conjecture is the following conjecture in number theory: :If :: A^x +B^y = C^z, :where ''A'', ''B'', ''C'', ''x'', ''y'', and ''z'' are positive integers with ''x'', ''y'', ''z'' ≥ 3, then ''A'', ''B'', and ''C'' have a common prime factor. Equivalently, :The equation A^x + B^y = C^z has no solutions in positive integers and pairwise coprime integers ''A, B, C'' if ''x, y, z'' ≥ 3. The conjecture was formulated in 1993 by Andrew Beal, a banker and amateur mathematician, while investigating generalizations of Fermat's Last Theorem. Since 1997, Beal has offered a monetary prize for a peer-reviewed proof of this conjecture or a counterexample. The value of the prize has increased several times and is currently $1 million. In some publications, this conjecture has occasionally been referred to as a generalized Fermat equation, the Mauldin conjecture, and the Tijdeman-Zagier conjecture. Related examples To illustrate, the solution 3^3 + 6^3 = 3^5 has bases with a c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Taxicab Number
In mathematics, the generalized taxicab number ''Taxicab''(''k'', ''j'', ''n'') is the smallest number — if it exists — that can be expressed as the sum of ''j'' ''k''th positive powers in ''n'' different ways. For ''k'' = 3 and ''j'' = 2, they coincide with taxicab numbers. :\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 — famously stated by Ramanujan. Euler showed that :\mathrm(4, 2, 2) = 635318657 = 59^4 + 158^4 = 133^4 + 134^4. However, ''Taxicab''(5, 2, ''n'') is not known for any ''n'' ≥ 2: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. The largest variable of \mathrm a^5+b^5=c^5+d^5 must be at least 3450. See also *Cabtaxi number In mathematics, the ''n''-th cabtaxi number, typically denoted Cabtaxi(''n''), is defined as the smallest positive int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Sum Of Powers Conjecture
Euler's conjecture is a disproved conjecture in mathematics related to Fermat's Last Theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers and greater than 1, if the sum of many th powers of positive integers is itself a th power, then is greater than or equal to : : ⇒ The conjecture represents an attempt to generalize Fermat's Last Theorem, which is the special case : if , then . Although the conjecture holds for the case (which follows from Fermat's Last Theorem for the third powers), it was disproved for and . It is unknown whether the conjecture fails or holds for any value . Background Euler was aware of the equality involving sums of four fourth powers; this, however, is not a counterexample because no term is isolated on one side of the equation. He also provided a complete solution to the four cubes problem as in Plato's number or the taxicab number 1729. The general solution of the equation :x_1^3+x_2^3=x_3^3+x_4^3 is :x_1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantine Equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents. Diophantine problems have fewer equations than unknowns and involve finding integers that solve simultaneously all equations. As such systems of equations define algebraic curves, algebraic surfaces, or, more generally, algebraic sets, their study is a part of algebraic geometry that is called '' Diophantine geometry''. The word ''Diophantine'' refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra. The mathematical study of Diophantine proble ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. 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 (AMS) is an association of p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
