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Sum Of Two Cubes
In mathematics, the sum of two cubes is a cubed number added to another cubed number. Factorization Every sum of cubes may be factored according to the identity a^3 + b^3 = (a + b)(a^2 - ab + b^2) in elementary algebra. Binomial numbers generalize this factorization to higher odd powers. Proof Starting with the expression, a^2-ab+b^2 and multiplying by (a+b)(a^2-ab+b^2) = a(a^2-ab+b^2) + b(a^2-ab+b^2). distributing ''a'' and ''b'' over a^2-ab+b^2, a^3 - a^2 b + ab^2 + a^2b - ab^2 + b^3 and canceling the like terms, a^3 + b^3. Similarly for the difference of cubes, \begin (a-b)(a^2+ab+b^2) & = a(a^2+ab+b^2) - b(a^2+ab+b^2) \\ & = a^3 + a^2 b + ab^2 \; - a^2b - ab^2 - b^3 \\ & = a^3 - b^3. \end "SOAP" mnemonic The mnemonic "SOAP", short for "Same, Opposite, Always Positive", helps recall of the signs: : Fermat's last theorem Fermat's last theorem in the case of exponent 3 states that the sum of two non-zero integer cubes does not result in a non-zero integer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sum And Difference Of 2 Cubes
Sum most commonly means the total of two or more numbers added together; see addition. Sum can also refer to: Mathematics * Sum (category theory), the generic concept of summation in mathematics * Sum, the result of summation, the addition of a sequence of numbers * 3SUM, a term from computational complexity theory * Band sum, a way of connecting mathematical knots * Connected sum, a way of gluing manifolds * Digit sum, in number theory * Direct sum, a combination of algebraic objects ** Direct sum of groups ** Direct sum of modules ** Direct sum of permutations ** Direct sum of topological groups * Einstein summation, a way of contracting tensor indices * Empty sum, a sum with no terms * Indefinite sum, the inverse of a finite difference * Kronecker sum, an operation considered a kind of addition for matrices * Matrix addition, in linear algebra * Minkowski addition, a sum of two subsets of a vector space * Power sum symmetric polynomial, in commutative algebra * Prefix sum, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonhard 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 Mathematical notation, 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 Kingdom of Prussia, Prussia. Euler is credited for popularizing the Greek letter \pi (lowercase Pi (letter), pi) to denote Pi, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aurifeuillean Factorization
In number theory, an aurifeuillean factorization, named after Léon-François-Antoine Aurifeuille, is factorization of certain integer values of the cyclotomic polynomials. Because cyclotomic polynomials are irreducible polynomials over the integers, such a factorization cannot come from an algebraic factorization of the polynomial. Nevertheless, certain families of integers coming from cyclotomic polynomials have factorizations given by formulas applying to the whole family, as in the examples below. Examples * Numbers of the form a^4 + 4b^4 have the following factorization ( Sophie Germain's identity): a^4 + 4b^4 = (a^2 - 2ab + 2b^2)\cdot (a^2 + 2ab + 2b^2). Setting a=1 and b=2^k, one obtains the following aurifeuillean factorization of \Phi_4(2^)=2^+1, where \Phi_4(x)=x^2+1 is the fourth cyclotomic polynomial: 2^+1 = (2^-2^+1)\cdot (2^+2^+1). * Numbers of the form a^6 + 27b^6 have the following factorization, where the first factor (a^2 + 3b^2) is the algebraic factorization o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sophie Germain's Identity
In mathematics, Sophie Germain's identity is a polynomial factorization named after Sophie Germain stating that \begin x^4 + 4y^4 &= \bigl((x + y)^2 + y^2\bigr)\cdot\bigl((x - y)^2 + y^2\bigr)\\ &= (x^2 + 2xy + 2y^2)\cdot(x^2 - 2xy + 2y^2). \end Beyond its use in elementary algebra, it can also be used in number theory to factorize integers of the special form x^4+4y^4, and it frequently forms the basis of problems in mathematics competitions. History Although the identity has been attributed to Sophie Germain, it does not appear in her works. Instead, in her works one can find the related identity \begin x^4+y^4 &= (x^2-y^2)^2+2(xy)^2\\ &= (x^2+y^2)^2-2(xy)^2.\\ \end Modifying this equation by multiplying y by \sqrt2 gives x^4+4y^4 = (x^2+2y^2)^2-4(xy)^2, a difference of two squares, from which Germain's identity follows. The inaccurate attribution of this identity to Germain was made by Leonard Eugene Dickson in his ''History of the Theory of Numbers'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Difference Of Two Squares
In elementary algebra, a difference of two squares is one squared number (the number multiplied by itself) subtracted from another squared number. Every difference of squares may be factored as the product of the sum of the two numbers and the difference of the two numbers: a^2-b^2 = (a+b)(a-b). In the reverse direction, the product of any two numbers can be expressed as the difference between the square of their average and the square of half their difference: xy = \left(\frac\right)^2 - \left(\frac\right)^2. Proof Algebraic proof The proof of the factorization identity is straightforward. Starting from the right-hand side, apply the distributive law to get (a+b)(a-b) = a^2+ba-ab-b^2. By the commutative law, the middle two terms cancel: ba - ab = 0 leaving (a+b)(a-b) = a^2-b^2. The resulting identity is one of the most commonly used in mathematics. Among many uses, it gives a simple proof of the AM–GM inequality in two variables. The proof holds not only for numbers ... [...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] |
Ramanujan 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-valued ... [...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] |
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases and have been known since antiquity to have infinitely many solutions.Singh, pp. 18–20 The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of ''Arithmetica''. Fermat added that he had a proof that was too large to fit in the margin. Although other statements claimed by Fermat without proof were subsequently proven by others and credited as theorems of Fermat (for example, Fermat's theorem on sums of two squares), Fermat's Last Theorem resisted proof, leading to doubt that Fermat ever had a correct proof. Consequently, the proposition became known as a conjecture rather than a theorem. After 358 years of effort by mathematicians, Wiles's proof of Fermat's Last Theorem, the first success ... [...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] |
Plus And Minus Signs
The plus sign () and the minus sign () are Glossary of mathematical symbols, mathematical symbols used to denote sign (mathematics), positive and sign (mathematics), negative functions, respectively. In addition, the symbol represents the operation of addition, which results in a Sum (mathematics), sum, while the symbol represents subtraction, resulting in a difference (mathematics), difference. Their use has been extended to many other meanings, more or less analogous. and are Latin terms meaning 'more' and 'less', respectively. The forms and are used in many countries around the world. Other designs include for plus and for minus. History Though the signs now seem as familiar as the alphabet or the Arabic numerals, they are not of great antiquity. The Egyptian hieroglyphic sign for addition, for example, resembles a pair of legs walking in the direction in which the text was written (Egyptian language, Egyptian could be written either from right to left or left to r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mnemonic
A mnemonic device ( ), memory trick or memory device is any learning technique that aids information retention or retrieval in the human memory, often by associating the information with something that is easier to remember. It makes use of elaborative encoding, retrieval cues and imagery as specific tools to encode information in a way that allows for efficient storage and retrieval. It aids original information in becoming associated with something more accessible or meaningful—which in turn provides better retention of the information. Commonly encountered mnemonics are often used for lists and in auditory system, auditory form such as Acrostic, short poems, acronyms, initialisms or memorable phrases. They can also be used for other types of information and in visual or kinesthetic forms. Their use is based on the observation that the human mind more easily remembers spatial, personal, surprising, physical, sexual, humorous and otherwise "relatable" information rather tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
