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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 tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantus
Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the author of a series of books called ''Arithmetica'', many of which are now lost. His texts deal with solving algebraic equations. Diophantine equations ("Diophantine geometry") and Diophantine approximations are important areas of mathematical research. Diophantus coined the term παρισότης (parisotes) to refer to an approximate equality. This term was rendered as ''adaequalitas'' in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. In modern use, Diophantine equations are usually algebraic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brahmagupta
Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the '' Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical treatise, and the '' Khaṇḍakhādyaka'' ("edible bite", dated 665), a more practical text. Brahmagupta was the first to give rules for computing with '' zero''. The texts composed by Brahmagupta were in elliptic verse in Sanskrit, as was common practice in Indian mathematics. As no proofs are given, it is not known how Brahmagupta's results were derived. In 628 CE, Brahmagupta first described gravity as an attractive force, and used the term "gurutvākarṣaṇam (गुरुत्वाकर्षणम्)" in Sanskrit to describe it. Life and career Brahmagupta was born in 598 CE according to his own statement. He lived in ''Bhillamāla'' in Gurjaradesa (modern Bhinmal in Rajasthan, India) during the reign of the Chavda dynasty ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Pell (mathematician)
John Pell (1 March 1611 – 12 December 1685) was an English mathematician and political agent abroad. Early life He was born at Southwick in Sussex. His father, also named John Pell, was from Southwick, and his mother was Mary Holland, from Halden in Kent. The second of two sons, Pell's older brother was Thomas Pell. By the time he was six, they were orphans, their father dying in 1616 and their mother the following year. John Pell the elder had a fine library, which proved valuable to the young Pell as he grew up. He was educated at Steyning Grammar School and entered Trinity College, Cambridge, at the age of 13. During his university career he became an accomplished linguist; even before taking a B.A. degree in 1629, he corresponded with Henry Briggs and other mathematicians. He was promoted by seniority to M.A. in 1630 and taught in the short-lived Chichester Academy set up by Samuel Hartlib. On 3 July 1632 he married Ithamaria Reginald (also rendered as Ithamara or Ithu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pell's Equation
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinates, the equation is represented by a hyperbola; solutions occur wherever the curve passes through a point whose ''x'' and ''y'' coordinates are both integers, such as the trivial solution with ''x'' = 1 and ''y'' = 0. Joseph Louis Lagrange proved that, as long as ''n'' is not a perfect square, Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accurately approximate the square root of ''n'' by rational numbers of the form ''x''/''y''. This equation was first studied extensively in India starting with Brahmagupta, who found an integer solution to 92x^2 + 1 = y^2 in his ''Brāhmasphuṭasiddhānta'' circa 628. Bhaskara II in the 12th century and Narayana Pandit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiles's Proof Of Fermat's Last Theorem
Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were almost universally considered inaccessible to prove by contemporaneous mathematicians, meaning that they were believed to be impossible to prove using current knowledge. Wiles first announced his proof on 23 June 1993 at a lecture in Cambridge entitled "Modular Forms, Elliptic Curves and Galois Representations". However, in September 1993 the proof was found to contain an error. One year later on 19 September 1994, in what he would call "the most important moment of isworking life", Wiles stumbled upon a revelation that allowed him to correct the proof to the satisfaction of the mathematical community. The corrected proof was published in 1995. Wiles's proof uses many techniques from algeb ... [...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 integers , , and satisfy the equation for any integer value of greater than 2. 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, the first successful proof was released in 1994 by Andrew Wiles and for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagorean Triple
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is one in which , and are coprime (that is, they have no common divisor larger than 1). For example, is a primitive Pythagorean triple whereas is not. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle. The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a^2+b^2=c^2; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides a=b=1 and c=\sqrt2 is a right triangle, but (1,1,\sqrt2) is not a Pythagorean triple because \sqrt2 is not an integer. Moreover, 1 and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy–Ramanujan Number
1729 is the natural number following 1728 and preceding 1730. It is a taxicab number, and is variously known as Ramanujan's number and the Ramanujan-Hardy number, after an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital. He related their conversation: The two different ways are: : 1729 = 13 + 123 = 93 + 103 The quotation is sometimes expressed using the term "positive cubes", since allowing negative perfect cubes (the cube of a negative integer) gives the smallest solution as 91 (which is a divisor of 1729; 1991 = 1729). :91 = 63 + (−5)3 = 43 + 33 Numbers that are the smallest number that can be expressed as the sum of two cubes in ''n'' distinct ways have been dubbed "taxicab numbers". The number was also found in one of Ramanujan's notebooks dated years before the incident, and was noted by Frénicle de Bessy in 1657. A commemorative plaque now appears at the site of the Ramanujan-Hardy inciden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Quadratic Equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not quadratic.) The numbers , , and are the '' coefficients'' of the equation and may be distinguished by respectively calling them, the ''quadratic coefficient'', the ''linear coefficient'' and the ''constant'' or ''free term''. The values of that satisfy the equation are called ''solutions'' of the equation, and '' roots'' or '' zeros'' of the expression on its left-hand side. A quadratic equation has at most two solutions. If there is only one solution, one says that it is a double root. If all the coefficients are real numbers, there are either two real solutions, or a single real double root, or two complex solutions that are complex conjugates of each other. A quadratic equation always has two roots, if complex roots are included; and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
