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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] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...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. *There are only finitely many positive integers that are not sums of ''distinct'' squares. The largest one is 128. The same applies for sums of distinct cubes (largest one is 12,758), distinct fourth powers (largest is 5,134,240), etc. See for a generalization to sums of polynomials. *Faulhaber's formula expresses 1^k + 2^k + 3^k + \cdots + n^k as a polynomial in , or alternatively in terms 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 posit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponentiation
In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, is the product (mathematics), product of multiplying bases: b^n = \underbrace_.In particular, b^1=b. The exponent is usually shown as a superscript to the right of the base as or in computer code as b^n. This binary operation is often read as " to the power "; it may also be referred to as " raised to the th power", "the th power of ", or, most briefly, " to the ". The above definition of b^n immediately implies several properties, in particular the multiplication rule:There are three common notations for multiplication: x\times y is most commonly used for explicit numbers and at a very elementary level; xy is most common when variable (mathematics), variables are used; x\cdot y is used for emphasizing that one ta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
1729 (number)
1729 is the natural number following 1728 and preceding 1730. It is the first nontrivial taxicab number, expressed as the sum of two cubic positive integers in two different ways. It is known as the Ramanujan number or Hardy–Ramanujan number after G. H. Hardy and Srinivasa Ramanujan. As a natural number 1729 is composite, the squarefree product of three prime numbers 7 × 13 × 19. It has as factors 1, 7, 13, 19, 91, 133, 247, and 1729. It is the third Carmichael number, and the first Chernick–Carmichael number. Furthermore, it is the first in the family of absolute Euler pseudoprimes, a subset of Carmichael numbers. 1729 is divisible by 19, the sum of its digits, making it a harshad number in base 10. 1729 is the dimension of the Fourier transform on which the fastest known algorithm for multiplying two numbers is based. This is an example of a galactic algorithm. 1729 can be expressed as the quadratic form. Investigating pairs of its distinct integer-valu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential discoveries in many other branches of mathematics, such as analytic number theory, complex analysis, and infinitesimal calculus. He also introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory. Euler has been called a "universal genius" who "was fully equipped with almost unlimited powers of imagination, intellectual gifts and extraordinary memory". He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. Euler is credited for popularizing the Greek letter \pi (lowercase pi) to denote the ratio of a circle's circumference to its diameter, as we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
