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Lucas Number
The Lucas sequence is an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci sequence. Individual numbers in the Lucas sequence are known as Lucas numbers. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences. The Lucas sequence has the same recursive relationship as the Fibonacci sequence, where each term is the sum of the two previous terms, but with different starting values. This produces a sequence where the ratios of successive terms approach the golden ratio, and in fact the terms themselves are roundings of integer powers of the golden ratio. The sequence also has a variety of relationships with the Fibonacci numbers, like the fact that adding any two Fibonacci numbers two terms apart in the Fibonacci sequence results in the Lucas number in between. The first few Lucas numbers are : 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cassini Identity
__notoc__ Cassini's identity (sometimes called Simson's identity) and Catalan's identity are mathematical identities for the Fibonacci numbers. Cassini's identity, a special case of Catalan's identity, states that for the ''n''th Fibonacci number, : F_F_ - F_n^2 = (-1)^n. Note here F_0 is taken to be 0, and F_1 is taken to be 1. Catalan's identity generalizes this: :F_n^2 - F_F_ = (-1)^F_r^2. Vajda's identity generalizes this: :F_F_ - F_F_ = (-1)^nF_F_. History Cassini's formula was discovered in 1680 by Giovanni Domenico Cassini, then director of the Paris Observatory, and independently proven by Robert Simson (1753). However Johannes Kepler presumably knew the identity already in 1608. Catalan's identity is named after Eugène Catalan (1814–1894). It can be found in one of his private research notes, entitled "Sur la série de Lamé" and dated October 1879. However, the identity did not appear in print until December 1886 as part of his collected works . This explains ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucas Sequence
In mathematics, the Lucas sequences U_n(P,Q) and V_n(P, Q) are certain constant-recursive integer sequences that satisfy the recurrence relation : x_n = P \cdot x_ - Q \cdot x_ where P and Q are fixed integers. Any sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences U_n(P, Q) and V_n(P, Q). More generally, Lucas sequences U_n(P, Q) and V_n(P, Q) represent sequences of polynomials in P and Q with integer coefficients. Famous examples of Lucas sequences include the Fibonacci numbers, Mersenne numbers, Pell numbers, Lucas numbers, Jacobsthal numbers, and a superset of Fermat numbers (see below). Lucas sequences are named after the French mathematician Édouard Lucas. Recurrence relations Given two integer parameters P and Q, the Lucas sequences of the first kind U_n(P,Q) and of the second kind V_n(P,Q) are defined by the recurrence relations: :\begin U_0(P,Q)&=0, \\ U_1(P,Q)&=1, \\ U_n(P,Q)&=P\cdot U_(P,Q)-Q\cdot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity (mathematics)
In mathematics, an identity is an equality (mathematics), equality relating one mathematical expression ''A'' to another mathematical expression ''B'', such that ''A'' and ''B'' (which might contain some variable (mathematics), variables) produce the same value for all values of the variables within a certain domain of discourse. In other words, ''A'' = ''B'' is an identity if ''A'' and ''B'' define the same function (mathematics), functions, and an identity is an equality between functions that are differently defined. For example, (a+b)^2 = a^2 + 2ab + b^2 and \cos^2\theta + \sin^2\theta =1 are identities. Identities are sometimes indicated by the triple bar symbol instead of , the equals sign. Formally, an identity is a universally quantified equality. Common identities Algebraic identities Certain identities, such as a+0=a and a+(-a)=0, form the basis of algebra, while other identities, such as (a+b)^2 = a^2 + 2ab +b^2 and a^2 - b^2 = (a+b)(a-b), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar example of modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in , but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is ''congruent'' to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12). Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss Congruence
In mathematics, Gauss congruence is a property held by certain sequences of integers, including the Lucas numbers and the divisor sum sequence. Sequences satisfying this property are also known as Dold sequences, Fermat sequences, Newton sequences, and realizable sequences. The property is named after Carl Friedrich Gauss (1777–1855), although Gauss never defined the property explicitly. Sequences satisfying Gauss congruence naturally occur in the study of topological dynamics, algebraic number theory and combinatorics. Definition A sequence of integers (a_1,a_2,\dots) satisfies Gauss congruence if : \sum_\mu(d)a_\equiv 0\pmod for every n\geq 1, where \mu is the Möbius function. By Möbius inversion, this condition is equivalent to the existence of a sequence of integers (b_1,b_2,\dots) such that : a_n=\sum_b_dd for every n\geq 1. Furthermore, this is equivalent to the existence of a sequence of integers (c_1,c_2,\dots) such that : a_n=c_1a_+c_2a_+\cdots+c_a_1+nc_n for eve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate (square Roots)
In mathematics, the conjugate of an expression of the form a + b \sqrt d is a - b \sqrt d, provided that \sqrt d does not appear in and . One says also that the two expressions are conjugate. In particular, the two solutions of a quadratic equation are conjugate, as per the \pm in the quadratic formula x = \frac. Complex conjugation is the special case where the square root is i = \sqrt, the imaginary unit. Properties As (a + b \sqrt d)(a - b \sqrt d) = a^2 - b^2 d and (a + b \sqrt d) + (a - b \sqrt d) = 2a, the sum and the product of conjugate expressions do not involve the square root anymore. This property is used for removing a square root from a denominator, by multiplying the numerator and the denominator of a fraction by the conjugate of the denominator (see Rationalisation). An example of this usage is: \frac = \frac = \frac. Hence: \frac = \frac{a^2 - db^2}. A corollary property is that the subtraction: :(a+b\sqrt d) - (a-b\sqrt d)= 2b\sqrt d, leaves only a term c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Fraction Decomposition
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator. The importance of the partial fraction decomposition lies in the fact that it provides algorithms for various computations with rational functions, including the explicit computation of antiderivatives, Taylor series expansions, inverse Z-transforms, and inverse Laplace transforms. The concept was discovered independently in 1702 by both Johann Bernoulli and Gottfried Leibniz. In symbols, the ''partial fraction decomposition'' of a rational fraction of the form \frac, where and are polynomials, is the expression of the rational fraction as \frac=p(x) + \sum_j \frac where is a polynomial, and, for each , the denominator is a power of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proof
A mathematical proof is a deductive reasoning, deductive Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning that establish logical certainty, to be distinguished from empirical evidence, empirical arguments or non-exhaustive inductive reasoning that establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Number
In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many writers begin the sequence with 0 and 1, although some authors start it from 1 and 1 and some (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the sequence begins : 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... The Fibonacci numbers were first described in Indian mathematics as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book . Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the ''Fibonacci Quarterly''. Appli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Equation
In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning is often used, where a ''functional equation'' is an equation that relates several values of the same function. For example, the logarithm functions are essentially characterized by the ''logarithmic functional equation'' \log(xy)=\log(x) + \log(y). If the domain of the unknown function is supposed to be the natural numbers, the function is generally viewed as a sequence, and, in this case, a functional equation (in the narrower meaning) is called a recurrence relation. Thus the term ''functional equation'' is used mainly for real functions and complex functions. Moreover a smoothness condition is often assumed for the solutions, since without such a condition, most functional equations have very irregular solutions. For example, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extension To Negative Integers
Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (proof theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * Extension (semantics), the set of things to which a property applies * Extension (simplicial set) * Extension by definitions * Extensional definition, a definition that enumerates every individual a term applies to * Extensionality Other uses * Extension of a function, defined on a larger domain * Extension of a polyhedron, in geometry * Extension of a line segment (finite) into an infinite line (e.g., extended base) * Exterior algebra, Grassmann's theory of extension, in geometry * Field extension, in Galois theory * Group extension, in abstract algebra and homological algebra * Homotopy extension property, in topology * Kolmogorov extension theorem, in probability theory * Linear extension, in order theory * Sh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Function
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably. The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed. Generating functions are sometimes called generating series, in that a series of terms can be said to be the generator of its sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
