|
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] |
List Of Disproved Mathematical Ideas
In mathematics, ideas are supposedly not accepted as fact until they have been rigorously proved. However, there have been some ideas that were fairly accepted in the past but which were subsequently shown to be false. This article is meant to serve as a repository for compiling a list of such ideas. *The idea of the Pythagoreans that all numbers can be expressed as a ratio of two whole numbers. This was disproved by one of Pythagoras' own disciples, Hippasus, who showed that the square root of two is what we today call an irrational number. One story claims that he was thrown off the ship in which he and some other Pythagoreans were sailing because his discovery was too heretical. *Fermat conjectured that all numbers of the form 2^+1 (what we call Fermat numbers) were prime. This, however, was disproved by Euler. *The idea that transcendental numbers were unusual. Disproved by Georg Cantor who showed that there are so many transcendental numbers that it is impossible to make ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plato Number
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution of higher learning on the European continent. Along with his teacher, Socrates, and his student, Aristotle, Plato is a central figure in the history of Ancient Greek philosophy and the Western and Middle Eastern philosophies descended from it. He has also shaped religion and spirituality. The so-called neoplatonism of his interpreter Plotinus greatly influenced both Christianity (through Church Fathers such as Augustine) and Islamic philosophy (through e.g. Al-Farabi). In modern times, Friedrich Nietzsche diagnosed Western culture as growing in the shadow of Plato (famously calling Christianity "Platonism for the masses"), while Alfred North Whitehead famously said: "the safest general characterization of the European philosophical t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantine Equations
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 problems that Di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Gerbicz
The name Robert is an ancient Germanic given name, from Proto-Germanic "fame" and "bright" (''Hrōþiberhtaz''). Compare Old Dutch ''Robrecht'' and Old High German ''Hrodebert'' (a compound of '' Hruod'' ( non, Hróðr) "fame, glory, honour, praise, renown" and ''berht'' "bright, light, shining"). It is the second most frequently used given name of ancient Germanic origin. It is also in use as a surname. Another commonly used form of the name is Rupert. After becoming widely used in Continental Europe it entered England in its Old French form ''Robert'', where an Old English cognate form (''Hrēodbēorht'', ''Hrodberht'', ''Hrēodbēorð'', ''Hrœdbœrð'', ''Hrœdberð'', ''Hrōðberχtŕ'') had existed before the Norman Conquest. The feminine version is Roberta. The Italian, Portuguese, and Spanish form is Roberto. Robert is also a common name in many Germanic languages, including English, German, Dutch, Norwegian, Swedish, Scots, Danish, and Icelandic. It can be use ... [...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'' ... [...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 integ ... [...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] |
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] |
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 + 52 ... [...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 exac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathologer
Burkard Polster (born 26 February 1965 in Würzburg) is a German mathematician who runs and presents the ''Mathologer'' channel on YouTube. Polster earned a doctorate from the University of Erlangen–Nuremberg in 1993 under the supervision of Karl Strambach. He works as an associate professor of mathematics at Monash University in Melbourne, Australia. Other universities that Polster has been affiliated with, before joining Monash in 2000, include the University of Würzburg, University at Albany, University of Kiel, University of California, Berkeley, University of Canterbury, and University of Adelaide. Polster's research involves topics in geometry, recreational mathematics, and the mathematics of everyday life, including how to tie shoelaces Shoelaces, also called shoestrings (US English) or bootlaces (UK English), are a system commonly used to secure shoes, boots, and other footwear. They typically consist of a pair of strings or cords, one for each shoe, finished off ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
YouTube
YouTube is a global online video platform, online video sharing and social media, social media platform headquartered in San Bruno, California. It was launched on February 14, 2005, by Steve Chen, Chad Hurley, and Jawed Karim. It is owned by Google, and is the List of most visited websites, second most visited website, after Google Search. YouTube has more than 2.5 billion monthly users who collectively watch more than one billion hours of videos each day. , videos were being uploaded at a rate of more than 500 hours of content per minute. In October 2006, YouTube was bought by Google for $1.65 billion. Google's ownership of YouTube expanded the site's business model, expanding from generating revenue from advertisements alone, to offering paid content such as movies and exclusive content produced by YouTube. It also offers YouTube Premium, a paid subscription option for watching content without ads. YouTube also approved creators to participate in Google's Google AdSens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
