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The Pippenger product is an unexpected Wallis-like formula for given by(1)(OEIS A084148 and A084149; Pippenger 1980). Here, the th term for is given by(2)(3)where is a double factorial and is the gamma function.

The value of the bit in a binary number. For the sequence of numbers 1, 2, 3, 4, ..., the least significant bits are therefore the alternating sequence 1, 0, 1, 0, 1, 0, ... (OEIS A000035). It can be represented as(1)(2)or(3)It is also given by the linear recurrenceequation(4)with (Wolfram 2002, p. 128).Analogously, the "most significant bit" is the value of the bit in an -bit representation.The least significant bit has Lambert series(5)where is a q-polygamma function.

In general, there is no unique matrix solution to the matrix equationEven in the case of parallel to , there are still multiple matrices that perform this transformation. For example, given , all the following matrices satisfy the above equation:Therefore, vector division cannot be uniquely defined in terms of matrices.However, if the vectors are represented by complex numbers or quaternions, vector division can be uniquely defined using the usual rules of complex division and quaternion algebra, respectively.

Following Yates (1980), a prime such that is a repeating decimal with decimal period shared with no other prime is called a unique prime. For example, 3, 11, 37, and 101 are unique primes, since they are the only primes with periods one (), two (), three (), and four () respectively. On the other hand, 41 and 271 both have period five, so neither is a unique prime.The unique primes are the primes such thatwhere is a cyclotomic polynomial, is the period of the unique prime, is the greatest common divisor, and is a positive integer.The first few unique primes are 3, 11, 37, 101, 9091, 9901, 333667, ... (OEIS A040017), which have periods 1, 2, 3, 4, 10, 12, 9, 14, 24, ... (OEIS A051627), respectively.

Pick any two relatively prime integers and , then the circle of radius centered at is known as a Ford circle. No matter what and how many s and s are picked, none of the Ford circles intersect (and all are tangent to the x-axis). This can be seen by examining the squared distance between the centers of the circles with and ,(1)Let be the sum of the radii(2)then(3)But , so and the distance between circle centers is the sum of the circle radii, with equality (and therefore tangency) iff . Ford circles are related to the Farey sequence (Conway and Guy 1996).If , , and are three consecutive terms in a Farey sequence, then the circles and are tangent at(4)and the circles and intersect in(5)Moreover, lies on the circumference of the semicircle with diameter and lies on the circumference of the semicircle with diameter (Apostol 1997, p. 101)...

A -automatic set is a set of integers whose base- representations form a regular language, i.e., a language accepted by a finite automaton or state machine. If bases and are incompatible (do not have a common power) and if an -automatic set and -automatic set are both of density 0 over the integers, then it is believed that is finite. However, this problem has not been settled.Some automatic sets, such as the 2-automatic consisting of numbers whose binary representations contain at most two 1s: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18, ... (OEIS A048645) have a simple arithmetic expression. However, this is not the case for general -automatic sets.

A number is said to be simply normal to base if its base- expansion has each digit appearing with average frequency tending to .A normal number is an irrational number for which any finite pattern of numbers occurs with the expected limiting frequency in the expansion in a given base (or all bases). For example, for a normal decimal number, each digit 0-9 would be expected to occur 1/10 of the time, each pair of digits 00-99 would be expected to occur 1/100 of the time, etc. A number that is normal in base- is often called -normal.A number that is -normal for every , 3, ... is said to be absolutely normal (Bailey and Crandall 2003).As stated by Kac (1959), "As is often the case, it is much easier to prove that an overwhelming majority of objects possess a certain property than to exhibit even one such object....It is quite difficult to exhibit a 'normal' number!" (Stoneham 1970).If a real number is -normal, then it is also -normal for and integers (Kuipers..

A real number that is -normal for every base 2, 3, 4, ... is said to be absolutely normal. As proved by Borel (1922, p. 198), almost all real numbers in are absolutely normal (Niven 1956, p. 103; Stoneham 1970; Kuipers and Niederreiter 1974, p. 71; Bailey and Crandall 2002).The first specific construction of an absolutely normal number was by Sierpiński (1917), with another method presented by Schmidt (1962). These results were both obtained by complex constructive devices (Stoneham 1970), and are by no means easy to construct (Stoneham 1970, Sierpiński and Schinzel 1988).

The extension ring obtained from a commutative unit ring (other than the trivial ring) when allowing division by all non-zero divisors. The ring of fractions of an integral domain is always a field.The term "ring of fractions" is sometimes used to denote any localization of a ring. The ring of fractions in the above meaning is then referred to as the total ring of fractions, and coincides with the localization with respect to the set of all non-zero divisors.When defining addition and multiplication of fractions, all that is required of the denominators is that they be multiplicatively closed, i.e., if , then ,(1)(2)Given a multiplicatively closed set in a ring , the ring of fractions is all elements of the form with and . Of course, it is required that and that fractions of the form and be considered equivalent. With the above definitions of addition and multiplication, this set forms a ring.The original ring may not embed in this ring of..

The division of two complex numbers can be accomplished by multiplying the numerator and denominator by the complex conjugate of the denominator, for example, with and , is given by(1)(2)(3)(4)(5)where denotes the complex conjugate. In component notation with ,(6)

Two complex numbers and are added together componentwise,In component form,(Krantz 1999, p. 1).

The word "base" in mathematics is used to refer to a particular mathematical object that is used as a building block. The most common uses are the related concepts of the number system whose digits are used to represent numbers and the number system in which logarithms are defined. It can also be used to refer to the bottom edge or surface of a geometric figure.A real number can be represented using any integer number as a base (sometimes also called a radix or scale). The choice of a base yields to a representation of numbers known as a number system. In base , the digits 0, 1, ..., are used (where, by convention, for bases larger than 10, the symbols A, B, C, ... are generally used as symbols representing the decimal numbers 10, 11, 12, ...).The digits of a number in base (for integer ) can be obtained in the Wolfram Language using IntegerDigits[x, b].Let the base representation of a number be written(1)(e.g., ). Then, for example, the number 10 is..

Archimedes' axiom, also known as the continuity axiom or Archimedes' lemma, survives in the writings of Eudoxus (Boyer and Merzbach 1991), but the term was first coined by the Austrian mathematician Otto Stolz (1883). It states that, given two magnitudes having a ratio, one can find a multiple of either which will exceed the other. This principle was the basis for the method of exhaustion, which Archimedes invented to solve problems of area and volume.Symbolically, the axiom states thatiff the appropriate one of following conditions is satisfied for integers and : 1. If , then . 2. If , then . 3. If , then . Formally, Archimedes' axiom states that if and are two line segments, then there exist a finite number of points , , ..., on such thatand is between and (Itô 1986, p. 611). A geometry in which Archimedes' lemma does not hold is called a non-Archimedean Geometry...

A binary plot of an integer sequence is a plot of the binary representations of successive terms where each term is represented as a column of bits with 1s colored black and 0s colored white. The columns are then placed side-by-side to yield an array of colored squares. Several examples are shown above for the positive integers , square numbers , Fibonacci numbers , and binomial coefficients .Binary plots can be extended to rational number sequences by placing the binary representations of numerators on top, and denominators on bottom, as illustrated above for the sequence .Similarly, by using other bases and coloring the base- digits differently, binary plots can be extended to n-ary plots.

The term "product" refers to the result of one or more multiplications. For example, the mathematical statement would be read " times equals ," where is the product.More generally, it is possible to take the product of many different kinds of mathematical objects, including those that are not numbers. For example, the product of two sets is given by the Cartesian product. In topology, the product of spaces can be defined by using the product topology. The product of two groups, vector spaces, or modules is given by the direct product. In category theory, the product of objects is given using the category product.The product symbol is defined by(1)Useful product identities include(2)(3)

The remainder obtained when dividing a polynomial by another polynomial . The polynomial remainder is implemented in the Wolfram Language as PolynomialRemainder[p, q, x], and is related to the polynomial quotient byFor example, the polynomial remainder of and is , corresponding to polynomial quotient .

The quotient of two polynomials and , discarding any polynomial remainder. Polynomial quotients are implemented in the Wolfram Language as PolynomialQuotient[p, q, x], and are related to the polynomial remainder byFor example, the polynomial quotient of and is , leaving remainder .

Synthetic division is a shortcut method for dividing two polynomials which can be used in place of the standard long division algorithm. This method reduces the dividend and divisor polynomials into a set of numeric values. After these values are processed, the resulting set of numeric outputs is used to construct the polynomial quotient and the polynomial remainder.For an example of synthetic division, consider dividing by . First, if a power of is missing from either polynomial, a term with that power and a zero coefficient must be inserted into the correct position in the respective polynomial. In this case the term is missing from the dividend while the term is missing from the divisor; therefore, is added between the quintic and the cubic terms of the dividend while is added between the cubic and the linear terms of the divisor:(1)and(2)respectively.Next, all the variables and their exponents () are removed from the dividend, leaving instead..

Ruffini's rule a shortcut method for dividing a polynomial by a linear factor of the form which can be used in place of the standard long division algorithm. This method reduces the polynomial and the linear factor into a set of numeric values. After these values are processed, the resulting set of numeric outputs is used to construct the polynomial quotient and the polynomial remainder.Note that Ruffini's rule is a special case of the more generalized notion of synthetic division in which the divisor polynomial is a monic linear polynomial. Confusingly, Ruffini's rule is sometimes referred to as synthetic division, thus leading to the common misconception that the scope of synthetic division is significantly smaller than that of the long division algorithm.For an example of Ruffini's rule, consider divided by . First, if a power of is missing from the dividend, a term with that power and a zero coefficient must be inserted into the correct position..

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