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Sixth Power
In arithmetic and algebra the sixth power of a number ''n'' is the result of multiplying six instances of ''n'' together. So: :. Sixth powers can be formed by multiplying a number by its fifth power, multiplying the square of a number by its fourth power, by cubing a square, or by squaring a cube. The sequence of sixth powers of integers is: :0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1000000, 1771561, 2985984, 4826809, 7529536, 11390625, 16777216, 24137569, 34012224, 47045881, 64000000, 85766121, 113379904, 148035889, 191102976, 244140625, 308915776, 387420489, 481890304, ... They include the significant decimal numbers 106 (a million), 1006 (a short-scale trillion and long-scale billion), 10006 (a long-scale trillion) and so on. Squares and cubes The sixth powers of integers can be characterized as the numbers that are simultaneously squares and cubes. In this way, they are analogous to two other classes of figurate numbers: the square triangular numbers, w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th century, Italian mathematician Giuseppe Peano formalized arithmetic with his Peano axioms, which are highly important to the field of mathematical logic today. History The prehistory of arithmetic is limited to a small number of artifacts, which may indicate the conception of addition and subtraction, the best-known being the Ishango bone from central Africa, dating from somewhere between 20,000 and 18,000 BC, although its interpretation is disputed. The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations: addition, subtraction, multiplication, and division, as early as 2000 BC. These artifacts do not always reveal the specific process used for solving problems, but t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lander, Parkin, And Selfridge Conjecture
The Lander, Parkin, and Selfridge conjecture concerns the integer solutions of equations which contain sums of like powers. The equations are generalisations of those considered in Fermat's Last Theorem. The conjecture is that if the sum of some ''k''-th powers equals the sum of some other ''k''-th powers, then the total number of terms in both sums combined must be at least ''k''. Background Diophantine equations, such as the integer version of the equation ''a''2 + ''b''2 = ''c''2 that appears in the Pythagorean theorem, have been studied for their integer solution properties for centuries. Fermat's Last Theorem states that for powers greater than 2, the equation ''a''''k'' + ''b''''k'' = ''c''''k'' has no solutions in non-zero integers ''a'', ''b'', ''c''. Extending the number of terms on either or both sides, and allowing for higher powers than 2, led to Leonhard Euler to propose in 1769 that for all integers ''n'' and ''k'' greater than 1, if the sum of ''n'' ''k''th powe ... [...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 problems that Di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Waring's Problem
In number theory, Waring's problem asks whether each natural number ''k'' has an associated positive integer ''s'' such that every natural number is the sum of at most ''s'' natural numbers raised to the power ''k''. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. Waring's problem was proposed in 1770 by Edward Waring, after whom it is named. Its affirmative answer, known as the Hilbert–Waring theorem, was provided by Hilbert in 1909. Waring's problem has its own Mathematics Subject Classification, 11P05, "Waring's problem and variants". Relationship with Lagrange's four-square theorem Long before Waring posed his problem, Diophantus had asked whether every positive integer could be represented as the sum of four perfect squares greater than or equal to zero. This question later became known as Bachet's conjecture, after the 1621 translation of Diophantus by Claude Gaspard Bachet de Méziriac, and it was solved by Joseph-Louis Lag ... [...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] |
Bhāskara II
Bhāskara II (c. 1114–1185), also known as Bhāskarāchārya ("Bhāskara, the teacher"), and as Bhāskara II to avoid confusion with Bhāskara I, was an Indian mathematician and astronomer. From verses, in his main work, Siddhānta Shiromani (सिद्धांतशिरोमणी), it can be inferred that he was born in 1114 in Vijjadavida (Vijjalavida) and living in the Sahyadri mountain ranges of Western Ghats, believed to be the town of Patan in Chalisgaon, located in present-day Khandesh region of Maharashtra by scholars. He is the only ancient mathematician who has been immortalized on a monument. In a temple in Maharashtra, an inscription supposedly created by his grandson Changadeva, lists Bhaskaracharya's ancestral lineage for several generations before him as well as two generations after him. Colebrooke who was the first European to translate (1817) Bhaskaracharya II's mathematical classics refers to the family as Maharashtrian Brahmins residing on the ban ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indian Mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, Bhaskara II, and Varāhamihira. The decimal number system in use today: "The measure of the genius of Indian civilisation, to which we owe our modern (number) system, is all the greater in that it was the only one in all history to have achieved this triumph. Some cultures succeeded, earlier than the Indian, in discovering one or at best two of the characteristics of this intellectual feat. But none of them managed to bring together into a complete and coherent system the necessary and sufficient conditions for a number-system with the same potential as our own." was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of zero as a number,: "...our decimal system, which (by t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Recorde
Robert Recorde () was an Anglo-Welsh physician and mathematician. He invented the equals sign (=) and also introduced the pre-existing plus and minus signs, plus sign (+) to English speakers in 1557. Biography Born around 1512, Robert Recorde was the second and last son of Thomas and Rose Recorde of Tenby, Pembrokeshire, in Wales. Recorde entered the University of Oxford about 1525, and was elected a Fellow of All Souls College, Oxford, All Souls College there in 1531. Having adopted medicine as a profession, he went to the University of Cambridge to take the degree of M.D. in 1545. He afterwards returned to Oxford, where he publicly taught mathematics, as he had done prior to going to Cambridge. He invented the "equals" sign. It appears that he afterwards went to London, and acted as physician to King Edward VI of England, Edward VI and to Mary I of England, Queen Mary, to whom some of his books are dedicated. He was also controller of the Royal Mint and served as Comptroll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Factor Exponent Notation
In his 1557 work ''The Whetstone of Witte'', British mathematician Robert Recorde proposed an exponent notation by prime factorisation, which remained in use up until the eighteenth century and acquired the name ''Arabic exponent notation''. The principle of Arabic exponents was quite similar to Egyptian fractions; large exponents were broken down into smaller prime numbers. Squares and cubes were so called; prime numbers from five onwards were called ''sursolids''. Although the terms used for defining exponents differed between authors and times, the general system was the primary exponent notation until René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ... devised the Cartesian exponent notation, which is still used today. This is a list of Recorde's terms. By co ... [...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 (e.g. ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface , or blackboard bold \mathbb. 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 number that is not rational is called irrational. Irrational numbers include , , , and . Since the set of rational numbers is countable, and the set of real numbers is uncountable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Louis J
Louis may refer to: * Louis (coin) * Louis (given name), origin and several individuals with this name * Louis (surname) * Louis (singer), Serbian singer * HMS Louis, HMS ''Louis'', two ships of the Royal Navy See also Derived or associated terms * Lewis (other) * Louie (other) * Luis (other) * Louise (other) * Louisville (other) * Louis Cruise Lines * Louis dressing, for salad * Louis Quinze, design style Associated names * * Chlodwig, the origin of the name Ludwig, which is translated to English as "Louis" * Ladislav and László - names sometimes erroneously associated with "Louis" * Ludovic, Ludwig (other), Ludwig, Ludwick, Ludwik, names sometimes translated to English as "Louis" {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
