* The ring of p-adic integers is the inverse limit of the rings Z / p < sup > n </ sup > Z ( see modular arithmetic ) with the index set being the natural numbers with the usual order, and the morphisms being " take remainder ".

In modular arithmetic, some numbers have a multiplicative inverse with respect to the modulus.

* In modular arithmetic, the modular additive inverse of is also defined: it is the number such that.

The multiplicative inverse has innumerable applications in algorithms of computer science, particularly those related to number theory, since many such algorithms rely heavily on the theory of modular arithmetic.

# Use the extended Euclidean algorithm to compute k < sup >− 1 </ sup >, the modular multiplicative inverse of k mod 2 < sup > w </ sup >, where w is the number of bits in a word.

In modular arithmetic, the modular multiplicative inverse of a is also defined: it is the number x such that ax ≡ 1 ( mod n ).

where N < sub > 1 </ sub >< sup >− 1 </ sup > denotes the modular multiplicative inverse of N < sub > 1 </ sub > modulo N < sub > 2 </ sub > and vice-versa for N < sub > 2 </ sub >< sup >− 1 </ sup >; the indices k < sub > a </ sub > and n < sub > a </ sub > run from 0 ,..., N < sub > a </ sub >− 1 ( for a

The Euclidean modular multiplicative inverse of 2 mod 3 is 2.

** simplify, use the modular multiplicative inverse if necessary

where is the modular multiplicative inverse of modulo.

# Ensure divides the order of by checking the existence of the following modular multiplicative inverse:,

:: Note that the notation does not denote the modular multiplication of times the modular multiplicative inverse of but rather the quotient of divided by, i. e., the largest integer value to satisfy the relation.

Modular exponentiation can be performed with a negative exponent e by finding the modular multiplicative inverse d of b modulo m using the extended Euclidean algorithm.

Image: Modular_pair. svg | In a modular lattice, the maps φ and ψ indicated by the arrows are mutually inverse isomorphisms.

The matrix will have an inverse if and only if its determinant is not zero, and does not have any common factors with the modular base.

In modular arithmetic the set of congruence classes relatively prime to the modulus n form a group under multiplication called the multiplicative group of integers modulo n. It is also called the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n. ( Units refers to elements with a multiplicative inverse.

Inversive congruential generators are a type of nonlinear congruential pseudorandom number generator, which use the modular multiplicative inverse ( if it exists ) to generate the next number in a sequence.

Elliptic modular functions can sometimes be expressed as the inverse functions of ratios of hypergeometric functions whose arguments a, b, c are 1, 1 / 2, 1 / 3, ... or 0.

In modular arithmetic, a branch of number theory, a primitive root modulo n is any number g with the property that any number coprime to n is congruent to a power of g modulo n. In other words, g is a generator of the multiplicative group of integers modulo n. That is, for every integer a coprime to n, there is an integer k such that g < sup > k </ sup > ≡ a ( mod n ).

Using modular exponentiation by repeated squaring, the running time of this algorithm is O ( k log < sup > 3 </ sup > n ), where k is the number of different values of a we test ; thus this is an efficient, polynomial-time algorithm.

A modular form of weight k for the group

: where a, b, c, d are integers satisfying ad − bc = 1, i. e., defines f to be a modular form of order k.

The second variable might be given in terms of the parameter m, or as the elliptic modulus k, where k < sup > 2 </ sup > = m, or in terms of the modular angle α, where m = sin < sup > 2 </ sup > α.

Given a modular form f ( z ) of weight k, the mth Hecke operator acts by the formula

The more general Ramanujan – Petersson conjecture for holomorphic cusp forms in the theory of elliptic modular forms for congruence subgroups has a similar formulation, with exponent ( k − 1 )/ 2 where k is the weight of the form.

Using fast algorithms for modular exponentiation, the running time of this algorithm is O ( k · log < sup > 3 </ sup > n ), where k is the number of different values of a we test.

In ordinary representation theory, the number of simple modules k ( G ) is equal to the number of conjugacy classes of G. In the modular case, the number l ( G ) of simple modules is equal to the number of conjugacy classes whose elements have order coprime to the relevant prime p, the so-called p-regular classes.

Then given integers a, b, c, d with ad − bc = 1 ( thus belonging to the modular group ), with c chosen so that c = kq for some integer k > 0, define

This ring is not commutative unless k equals the field ( see modular arithmetic ).

* The ( 3-dimensional ) metaplectic group is a double cover of SL < sub > 2 </ sub >( R ) playing an important role in the theory of modular forms.

As a result WSF files provide a means for code reuse: one can write a library of classes or functions in one or more < tt >. vbs </ tt > files, and include those files in one or more WSF files to use and reuse that functionality in a modular way.

Fermat's little theorem states that if p is a prime number, then for any integer a, the number a < sup > p </ sup > − a is an integer multiple of p. In the notation of modular arithmetic, this says

There exist efficient algorithms for computing the modular exponentiations h < sup >( p – 1 )/ q </ sup > mod p and g < sup > x </ sup > mod p, such as exponentiation by squaring.

In number theory, the theory of Hilbert modular forms is concerned with the study of certain functions on the direct product H < sup > n </ sup > of n copies of the upper half-plane.

Yet another space interesting to number theorists is the Siegel upper half-space H < sub > n </ sub >, which is the domain of Siegel modular forms.

Left to right, modular connectors: < ul > < li > eight-contact 8P8C plug ( used for RJ45, RJ49, RJ61 and others )</ li > < li > six-contact 6P6C plug used for RJ25 </ li > < li > four-contact 6P4C plug used for RJ14 ( often also used instead of 6P2C for RJ11 )</ li > < li > four-contact 4P4C handset plug ( also popularly, though incorrectly, called " RJ9 ", " RJ10 ", or " RJ22 ")</ li > </ ul > RJ11, RJ14, and RJ25 can be plugged into the same six-pin 6P6C jack, pictured.

* The modular discriminant Δ ( τ ) is proportional to the 24th power of the Dedekind eta function η ( τ ): Δ ( τ ) = ( 2π )< sup > 12 </ sup > η ( τ )< sup > 24 </ sup >.

The two transformations τ → τ + 1 and τ → τ < sup >− 1 </ sup > together generate a group called the modular group, which we may identify with the projective special linear group.

If q = e < sup > 2πiτ </ sup >, then q < sup >− 1 / 60 </ sup > G ( q ) and q < sup > 11 / 60 </ sup > H ( q ) are modular functions of τ.

That is, for every prime number p with, one has the modular arithmetic relations that either or: the final digits is a 1 or a 5.

NeXT's second attempt came in 1994 with the Enterprise Objects Framework ( EOF ) version 1, a complete rewrite that was far more modular and OpenStep compatible.

A holomorphic function in the upper half plane which is invariant under linear fractional transformations with integer coefficients and determinant 1 is called a modular form.

Therefore once the role of some low-dimensional Lie groups such as GL ( 2 ) in the theory of modular forms had been recognised, and with hindsight GL ( 1 ) in class field theory, the way was open at least to speculation about GL ( n ) for general n > 2.

Certain curves had been known to be both elliptic curves ( of genus 1 ) and modular curves, before the conjecture was formulated ( about 1955 ).

The surprising part of the conjecture was the extension to factors of Jacobians of modular curves of genus > 1.

* lowest power modular transmitter: generally 1. 0 kW, using 600 W modules.

* highest power modular transmitter: 1. 0 MW ( for LW, MW ).

It is made with modular raised square textured granite slabs ( each slab costing approximately $ 1, 500: $ 1, 000 materials plus $ 500 labour ), features a diagonally running zinc canopy along the northern hypotenuse of the " square ", a movable plinth which serves as a stage for concerts and other performances, a row of lighted fountains set directly into the pavement, a row of small trees along the southern edge, a transparent canopy over the plinth, and a new entrance to Dundas subway station below.

RJ11, RJ14, and RJ25 all use the same six-position modular connector, thus are physically identical except for the different number of contacts ( two, four and six respectively ) allowing connections for 1, 2 or 3 phone lines respectively.

An important class of congruence subgroups is given by reduction of the ring of entries: in general given a group such as the special linear group SL ( n, Z ) we can reduce the entries to modular arithmetic in Z / NZ for any N > 1, which gives a homomorphism

The Clinchfield layout measured and was noteworthy for a number of reasons: 1 ) it was highly portable because of its modular construction, 2 ) showed the potential of high scenery to track ratios possible in 1: 160, 3 ) used a unique aluminum frame and ( at the time revolutionary ) styrofoam construction to cut down on weight and 4 ) was highly prototypical for the era.