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Page "Integer factorization" ¶ 16
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equivalently and is
# As exploratory data analysis, an ANOVA is an organization of an additive data decomposition, and its sums of squares indicate the variance of each component of the decomposition ( or, equivalently, each set of terms of a linear model ).
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients ( or equivalently — by clearing denominators — with integer coefficients ).
is an Icelandic ( and equivalently Old Norse ) term consisting of two parts.
Similarly, P is said to satisfy the descending chain condition ( DCC ) if every descending chain of elements eventually terminates, or equivalently if any descending sequence
Inflation is defined either as the devaluation of a currency or equivalently the rise of prices relative to a currency.
To establish that a function is computable by Turing machine, it is usually considered sufficient to give an informal English description of how the function can be effectively computed, and then conclude " By the Church – Turing thesis " that the function is Turing computable ( equivalently partial recursive ).
One reason for the popularity of the dissociation constant in biochemistry and pharmacology is that in the frequently encountered case where x = y = 1, K < sub > d </ sub > has a simple physical interpretation: when = K < sub > d </ sub >, = or equivalently /(+)= 1 / 2.
Because of this, it is traditional to define the decision problem equivalently as: the set of inputs for which the problem returns yes.
The depth of the study of general Diophantine equations is shown by the characterisation of Diophantine sets as equivalently described as recursively enumerable.
or equivalently, integrating by parts, assuming ƒ < sup >( 2p )</ sup > is differentiable again and recalling that the odd Bernoulli numbers are zero:
Note that this derivation does assume that ƒ ( x ) is sufficiently differentiable and well-behaved ; specifically, that ƒ may be approximated by polynomials ; equivalently, that ƒ is a real analytic function of exponential type.
Define f ( x ) to be the highest power of the maximal ideal M containing x ( equivalently, to the power of the generator of the maximal ideal that x is associated to ).
In abstract algebra, a field is a commutative ring which contains a multiplicative inverse for every nonzero element, equivalently a ring whose nonzero elements form an abelian group under multiplication.
For example, an ester ( RCOOR ') has an ester functional group ( COOR ) and is composed of an alkoxy moiety (- OR ') and an acyl moiety ( RCO -), or, equivalently, it may be divided into carboxylate ( RCOO -) and alkyl (- R ') moieties.
Matching the theory's prediction to observational results for planetary orbits ( or, equivalently, assuring that the weak-gravity, low-speed limit is Newtonian mechanics ), the proportionality constant can be fixed as κ = 8πG / c < sup > 4 </ sup >, with G the gravitational constant and c the speed of light.
This is not a hypothesis but an illustration of the fact that any description of the physical world can be equivalently expressed in more than one way.
or, equivalently, the Wirtinger derivative of ƒ with respect to the complex conjugate of z is zero:
Under Galilean transformations, the time t < sub > 2 </ sub > − t < sub > 1 </ sub > between two events is the same for all inertial reference frames and the distance between two simultaneous events ( or, equivalently, the length of any object, | r < sub > 2 </ sub > − r < sub > 1 </ sub >|) is also the same.
The logistic map is a polynomial mapping ( equivalently, recurrence relation ) of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations.

equivalently and N
momenta ; SI unit kg m / s, or, equivalently, N s ) is the product of the mass and velocity of an object.
If μ is not a positive measure, then N is μ-null if N is | μ |- null, where | μ | is the total variation of μ ; equivalently, if every measurable subset A of N satisfies μ ( A )
The electric field is a vector field with SI units of newtons per coulomb ( N C < sup >− 1 </ sup >) or, equivalently, volts per metre ( V m < sup >− 1 </ sup >).
Formally, the discrete cosine transform is a linear, invertible function ( where denotes the set of real numbers ), or equivalently an invertible N × N square matrix.
Formally, the discrete sine transform is a linear, invertible function F: R < sup > N </ sup > < tt >-></ tt > R < sup > N </ sup > ( where R denotes the set of real numbers ), or equivalently an N × N square matrix.
In the SI system, the unit is the pascal ( Pa ) ( or a multiple thereof, often megapascals ( MPa ), using the mega-prefix ); or, equivalently to pascals, newtons per square metre ( N / m² ).
The formula predicts the reliability of a new test composed by replicating the current test N times ( or, equivalently, creating a test with N parallel forms of the current exam ).
where I ranges through the non-zero ideals of the ring of integers O < sub > K </ sub > of K and N < sub > K / Q </ sub >( I ) denotes the absolute norm of I ( which is equal to both the index of I in O < sub > K </ sub > or equivalently the cardinality of quotient ring O < sub > K </ sub > / I ).
The Bingham distribution is a distribution over axes in N dimensions, or equivalently, over points on the ( N − 1 )- dimensional sphere with the antipodes identified.
The power of a point can be defined equivalently as the product of distances from the point P to the two intersection points of any ray emanating from P. For example, in Figure 1, a ray emanating from P intersects the circle in two points, M and N, whereas a tangent ray intersects the circle in one point T ; the horizontal ray from P intersects the circle at A and B, the endpoints of the diameter.
With the normalization used in the previous section, G ( 0 ) gives the mean number of diffusers in the volume < N >, or equivalently — with knowledge of the observation volume size — the mean concentration:

equivalently and prime
This is really the same construction, as the Stone space of this Boolean algebra is the set of ultrafilters ( or equivalently prime ideals, or homomorphisms to the 2 element Boolean algebra ) of the Boolean algebra, which is the same as the set of ultrafilters on X.
A Sophie Germain prime p > 3 is of the form 6k − 1 or, equivalently, p ≡ 5 ( mod 6 ) — as is its matching safe prime 2p + 1.
We note that the other form for a prime p > 3 is 6k + 1 or, equivalently, p ≡ 1 ( mod 6 ), and that 3 |( 2p + 1 ) — thus excluding such p from the Sophie Germain prime domain.
If n is a positive integer, the integers between 1 and n − 1 which are coprime to n ( or equivalently, the congruence classes coprime to n ) form a group with multiplication modulo n as the operation ; it is denoted by Z < sub > n </ sub >< sup >×</ sup > and is called the group of units modulo n or the group of primitive classes modulo n. As explained in the article multiplicative group of integers modulo n, this group is cyclic if and only if n is equal to 2, 4, p < sup > k </ sup >, or 2 p < sup > k </ sup > where p < sup > k </ sup > is a power of an odd prime number.
* An affine algebraic set V is a variety if and only if I ( V ) is a prime ideal ; equivalently, V is a variety if and only if its coordinate ring is an integral domain.
With the exception of 7, a safe prime q is of the form 6k − 1 or, equivalently, q ≡ 5 ( mod 6 ) — as is p > 3 ( c. f.
Similarly, with the exception of 5, a safe prime q is of the form 4k − 1 or, equivalently, q ≡ 3 ( mod 4 ) — trivially true since ( q − 1 ) / 2 must evaluate to an odd natural number.
Combining both forms using lcm ( 6, 4 ) we determine that a safe prime q > 7 also must be of the form 12k − 1 or, equivalently, q ≡ 11 ( mod 12 ).
Finally, let p be a non-zero prime ideal in A, or equivalently, a maximal ideal, so that the residue A / p is a field.
Every prime power ( except powers of 2 ) has a primitive root ; thus the multiplicative group of integers modulo p < sup > n </ sup > ( or equivalently, the unit group of the ring ) is cyclic.
that is irreducible ( or equivalently prime ) in the ring-theoretic sense: its only Eisenstein divisors are the units (± 1, ± ω, ± ω < sup > 2 </ sup >), a + bω itself and its associates.
A Giuga number is a composite number n such that for each of its distinct prime factors p < sub > i </ sub > we have, or equivalently such that for each of its distinct prime factors p < sub > i </ sub > we have.

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