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Congruum
In number theory, a congruum (plural ''congrua'') is the difference between successive square numbers in an arithmetic progression of three squares. The congruum problem is the problem of finding squares in arithmetic progression and their associated congrua. It can be formalized as a Diophantine equation. Fibonacci solved the congruum problem by finding a parameterized formula for generating all congrua, together with their associated arithmetic progressions. According to this formula, each congruum is four times the area of a Pythagorean triangle, a right triangle whose sides are integers. Congrua are also closely connected with congruent numbers, the areas of right triangles whose sides are rational numbers. Every congruum is a congruent number, and every congruent number is a congruum multiplied by the square of a rational number. Fibonacci claimed without proof that it is impossible for a congruum to be a square number. This was later proven by Pierre de Fermat as Fermat's r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagorean Triangle
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A triangle whose side lengths are a Pythagorean triple is a right triangle and called a Pythagorean triangle. 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. Every Pythagorean triple can be scaled to a unique primitive Pythagorean triple by dividing by their greatest common divisor. Conversely, every Pythagorean triple can be obtained by multiplying the elements of a primitive Pythagorean triple by a positive integer (the same for the three elements). 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 s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automedian Triangle
In plane geometry, an automedian triangle is a triangle in which the lengths of the three medians (the line segments connecting each vertex to the midpoint of the opposite side) are proportional to the lengths of the three sides, in a different order. The three medians of an automedian triangle may be translated to form the sides of a second triangle that is similar to the first one. Characterization The side lengths of an automedian triangle satisfy the formula 2a^2=b^2+c^2 or a permutation thereof, analogous to the Pythagorean theorem characterizing right triangles as the triangles satisfying the formula a^2+b^2=c^2. Equivalently, in order for the three numbers a, b, and c to be the sides of an automedian triangle, the sequence of three squared side lengths b^2, a^2, and c^2 should form an arithmetic progression.. That is, b^2-k = a^2, and a^2-k = c^2 (for example, if b=17, a=13, and c=7, then k=120: 17^2-120 = 13^2 = 169, and 13^2-120 = 7^2 = 49). Construction from right tria ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Book Of Squares
''The Book of Squares'', ''(Liber Quadratorum'' in the original Latin) is a book on algebra by Leonardo Fibonacci, published in 1225. It was dedicated to Frederick II, Holy Roman Emperor. The ''Liber quadratorum'' has been passed down by a single 15th-century manuscript, the so-called ms. ''E 75 Sup.'' of the Biblioteca Ambrosiana (Milan, Italy), ff. 19r–39v. During the 19th century, the work was published for the first time in a printed edition by Baldassarre Boncompagni Ludovisi, prince of Piombino. Appearing in the book is Fibonacci's identity, establishing that the set of all sums of two squares is closed under multiplication. The book anticipated the works of later mathematicians such as Fermat and Euler. The book examines several topics in number theory, among them an inductive method for finding Pythagorean triples based on the sequence of odd integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive nat ... [...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] |
Fibonacci
Leonardo Bonacci ( – ), commonly known as Fibonacci, was an Italians, Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, ''Fibonacci'', is first found in a modern source in a 1838 text by the Franco-Italian mathematician Guglielmo Libri Carucci dalla Sommaja, Guglielmo Libri and is short for ('son of Bonacci'). However, even as early as 1506, Perizolo, a notary of the Holy Roman Empire, mentions him as "Lionardo Fibonacci". Fibonacci popularized the Hindu–Arabic numeral system, Indo–Arabic numeral system in the Western world primarily through his composition in 1202 of (''Book of Calculation'') and also introduced Europe to the sequence of Fibonacci numbers, which he used as an example in . Biography Fibonacci was born around 1170 to Guglielmo, an Italian merchant and customs official who directed a trading post in Béjaïa, Bugia, modern-day Béjaïa, Algeria ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Sequences
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description . The sequence 0, 3, 8, 15, ... is formed according to the formula ''n''2 − 1 for the ''n''th term: an explicit definition. Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number, , even though we do not have a formula for the ''n''th perfect number. Computable and definable sequences An integer sequence is computable if there exists an algorithm that, given '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantine Equations
''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name: *Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated ... * Diophantine equation * Diophantine quintuple * Diophantine set {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spiral Of Theodorus
In geometry, the spiral of Theodorus (also called the square root spiral, Pythagorean spiral, or Pythagoras's snail) is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyrene. Construction The spiral is started with an isosceles right triangle, with each leg having unit length. Another right triangle (which is the ''only'' automedian right triangle) is formed, with one leg being the hypotenuse of the prior right triangle (with length the square root of 2) and the other leg having length of 1; the length of the hypotenuse of this second right triangle is the square root of 3. The process then repeats; the nth triangle in the sequence is a right triangle with the side lengths \sqrt and 1, and with hypotenuse \sqrt. For example, the 16th triangle has sides measuring 4=\sqrt, 1 and hypotenuse of \sqrt. History and uses Although all of Theodorus' work has been lost, Plato put Theodorus into his dialogue '' Theaetetus'', which tells of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared". The name ''square'' number comes from the name of the shape. The unit of area is defined as the area of a unit square (). Hence, a square with side length has area . If a square number is represented by ''n'' points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of ''n''; thus, square numbers are a type of Figurate number, figurate numbers (other examples being Cube (algebra), cube numbers and triangular numbers). In the Real number, real number system, square numbers are non-negative. A non-negative integer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birch And Swinnerton-Dyer Conjecture
In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. It is named after mathematicians Bryan John Birch and Peter Swinnerton-Dyer, who developed the conjecture during the first half of the 1960s with the help of machine computation. Only special cases of the conjecture have been proven. The modern formulation of the conjecture relates to arithmetic data associated with an elliptic curve ''E'' over a number field ''K'' to the behaviour of the Hasse–Weil ''L''-function ''L''(''E'', ''s'') of ''E'' at ''s'' = 1. More specifically, it is conjectured that the rank of the abelian group ''E''(''K'') of points of ''E'' is the order of the zero of ''L''(''E'', ''s'') at ''s'' = 1. The first non-ze ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
