In number theory, the Legendre symbol is a multiplicative function with values 1, − 1, 0 that is a quadratic character modulo a prime number p: its value on a ( nonzero ) quadratic residue mod p is 1 and on a quadratic non-residue is − 1.

One obtains the value f ( r ) by substitution of the value r for the symbol X in P. One reason to distinguish between polynomials and polynomial functions is that over some rings different polynomials may give rise to the same polynomial function ( see Fermat's little theorem for an example where R is the integers modulo p ).

Usually, the modulo function maps any integer modulo N to one of the numbers 0, 1, 2, ...,, where.

It turns out that ,< sup > 14 </ sup > at least formally ( modulo such issues as the convergence of the sum ), for every choice of the billiard ball's initial, nonrelativistic wave function before the Cauchy horizon, such a sum over histories produces unique, self-consistent probabilities for the outcomes of all sets of subsequent measurements.

The totient function is important mainly because it gives the order of the multiplicative group of integers modulo n ( the group of units of the ring ).

Most early signature schemes were of a similar type: they involve the use of a trapdoor permutation, such as the RSA function, or in the case of the Rabin signature scheme, computing square modulo composite n. A trapdoor permutation family is a family of permutations, specified by a parameter, that is easy to compute in the forward direction, but is difficult to compute in the reverse direction without already knowing the private key.

* Programming example for the modulo function

In the opposite direction, given a group homomorphism on the unit group modulo k, we can lift to a completely multiplicative function on integers relatively prime to k and then extend this function to all integers by defining it to be 0 on integers having a non-trivial factor in common with k. The resulting function will then be a Dirichlet character.

The order of ( i. e. the number of elements in ) Z < sub > n </ sub >< sup >×</ sup > is given by Euler's totient function Euler's theorem says that a < sup > φ ( n )</ sup > ≡ 1 ( mod n ) for every a coprime to n ; the lowest power of a which is congruent to 1 modulo n is called the multiplicative order of a modulo n. In particular, for a to be a primitive root modulo n, φ ( n ) has to be the smallest power of a which is congruent to 1 modulo n.

Since 4 and 6 are congruent modulo 2, a function defined on the integers modulo 2 must give the same output when the input is 6 that it gives when the input is 4.

The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n. This is the group of units of the ring Z < sub > n </ sub >; it has φ ( n ) elements, φ being Euler's totient function, and is denoted as U ( n ) or U ( Z < sub > n </ sub >).

Here o < sub > r </ sub >( n ) is the multiplicative order of n modulo r, log is the binary logarithm, and is Euler's totient function of r.

An L-function L ( E, s ) can be defined for an elliptic curve E by constructing an Euler product from the number of points on the curve modulo each prime p. This L-function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined for a binary quadratic form.

Its round function is very simple: add a 32-bit subkey modulo 2 < sup > 32 </ sup >, put the result through a layer of S-boxes, and rotate that result left by 11 bits.

Thus, the function of antioxidant systems is not to remove oxidants entirely, but instead to keep them at an optimum level.

If f is not a function, but is instead a partial function, it is called a partial operation.

where is the Boltzmann constant, T is temperature ( assumed to be a well-defined quantity ), is the degeneracy ( meaning, the number of levels having energy ; sometimes, the more general ' states ' are used instead of levels, to avoid using degeneracy in the equation ), N is the total number of particles and Z ( T ) is the partition function.

There is no " function " keyword ; instead, a function is indicated by the parentheses of an argument list.

Interpretations of crannog function are not static through time ; instead they appear to change in both the archaeological and historic records.

Since such life utilised strong nuclear forces instead of electromagnetic interactions, it was posited that life might function millions of times faster than typical on Earth.

It was Euler ( presumably around 1740 ) who turned his attention to the exponential function instead of logarithms, and obtained the correct formula now named after him.

* An equalization filter is not designed to fully pass or block any frequency, but instead to gradually vary the amplitude response as a function of frequency: filters used as pre-emphasis filters, equalizers, or tone controls are good examples.

Hyperbolas arise in practice in many ways: as the curve representing the function in the Cartesian plane, as the appearance of a circle viewed from within it, as the path followed by the shadow of the tip of a sundial, as the shape of an open orbit ( as distinct from a closed and hence elliptical orbit ), such as the orbit of a spacecraft during a gravity assisted swing-by of a planet or more generally any spacecraft exceeding the escape velocity of the nearest planet, as the path of a single-apparition comet ( one travelling too fast to ever return to the solar system ), as the scattering trajectory of a subatomic particle ( acted on by repulsive instead of attractive forces but the principle is the same ), and so on.

As a result, the continuous eigenstates | x ⟩ are normalized to the delta function instead of unity:

He also often uses linguistic terms for Perl language constructs, so instead of traditional terms such as " variable ", " function ", and " accessor " he sometimes says " noun ", " verb ", and " topicalizer ".

Alternatively, under penultimate hop popping this function may instead be performed by the LSR directly connected to the LER.

A security reduction for the protocol is given in the case where, instead of a hash function, a random oracle answers each query randomly but consistently ; the oracle is assumed to be available to all parties including the attacker, as the hash function is.

Using λ instead of φ ( n ) allows more choices for d. λ can also be defined using the Carmichael function, λ ( n ).

This is different from conventional models that center on individuals, structures, departments and units separate in part from the whole instead of recognizing the interdependence between groups of individuals, structures and processes that enable an organization to function.

Second, OS-9 does not have a Unix-style fork () system call — instead it has a system call which creates a process running a specified program, performing much the same function as a fork-exec or a spawn.

A backward difference uses the function values at x and x − h, instead of the values at x + h and x:

# Using a mathematical expression, such as a polynomial or a trigonometric function, and a single point on the corresponding curve instead of storing or transmitting the entire graphic curve or a series of points on it.

If the values of the function lie in an infinite-dimensional vector space instead of R or C, usually other definitions of measurability are used, such as weak measurability and Bochner measurability.

The use of the probability density instead of a probability in specifying the likelihood function above may be justified in a simple way.

If, instead, the m letters of the alphabet are mapped to, then the encryption and decryption function for the Atbash cipher becomes:

An example of an arithmetic function is the non-principal character ( mod 4 ) defined by

For such data, a function that extracts the numeric part k of the file name and returns k mod n would be nearly optimal.

1, Euler's theorem says that a < sup > φ ( k )</ sup > ≡ 1 ( mod k ) ( where φ ( k ) is the totient function ).

Ramanujan's tau function τ and the divisor function σ < sub > 11 </ sub > are related by the remarkable congruence τ ( n ) ≡ σ < sub > 11 </ sub >( n ) ( mod 691 ).

x < sup > d </ sup > r mod N. Finally it is unblinded using the function D ( z )

A quantitative form of Dirichlet's theorem states that if N ≥ 2 is an integer and a is coprime to N, then the proportion of the primes p congruent to a mod N is asymptotic to 1 / n, where n = φ ( N ) is the Euler totient function.

The quadratic sieve attempts to find pairs of integers x and y ( x ) ( where y ( x ) is a function of x ) satisfying a much weaker condition than x < sup > 2 </ sup > ≡ y < sup > 2 </ sup > ( mod n ).

Unfortunately, the way most computer languages implement the remainder function, − 2 mod 7 returns a result of-2.

If a set ' S ' is ill-distributed modulo p ( by virtue, for example, of being excluded from the congruence classes A < sub > p </ sub >) then the Fourier coefficients of the characteristic function f < sub > p </ sub > of the set S mod p are in average large.

Consider the linear probing hash function h ( k, i ): ( h `( k ) + i ) mod N. With k being the key, i the probing-iteration, N being the number of slots in the hash-table and h `( k ) being the secondary-hash function.