Euclid poses the problem: " Given two numbers not prime to one another, to find their greatest common measure ".

Given a general algorithm for integer factorization, one can factor any integer down to its constituent prime factors by repeated application of this algorithm.

Without doubt, the single most significant paper concerning the distribution of prime numbers was Riemann's 1859 memoir On the Number of Primes Less Than a Given Magnitude, the only paper he ever wrote on the subject.

Bernhard Riemann in his memoir " On the Number of Primes Less Than a Given Magnitude " published in 1859 extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation and established a relation between its zeros and the distribution of prime numbers.

Informally it solves the following problem: Given an integer N, find its prime factors.

Given a homogeneous prime ideal P of, let X be a subset of P < sup > n </ sup >( k ) consisting of all roots of polynomials in P .< ref > The definition makes sense since if and only if for any nonzero λ in k .</ ref > Here we show X admits a structure of variety by showing locally it is an affine variety.

The prime mechanism then appeared to be this: Given a space X carrying a vector bundle, that implied in the homotopy category a mapping from X to a classifying space BG, for the relevant linear group G. For the homotopy theory the relevant information is carried by compact subgroups such as the orthogonal groups and unitary groups of G. Once the cohomology H *( BG ) was calculated, once and for all, the contravariance property of cohomology meant that characteristic classes for the bundle would be defined in H *( X ) in the same dimensions.

Given a polynomial equation with rational coefficients, if it has rational solution, then this also yields a real solution and a p-adic solution, as the rationals embed in the reals and p-adics: a global solution yields local solutions at each prime.

Given a prime spot at a comedy club, he bombs after clashing with both Doris and Montgomery over his new lifestyle.

Given the specific case of N being the product of distinct odd prime numbers p and q, the structure of the squaring map:

# Given any two reduced alternating diagrams D < sub > 1 </ sub > and D < sub > 2 </ sub > of an oriented, prime alternating link: D < sub > 1 </ sub > may be transformed to D < sub > 2 </ sub > by means of a sequence of certain simple moves called flypes.

Given an arithmetic function and a prime, define the formal power series, called the Bell series of modulo as:

Given the intrinsic unpredictability of the timing and trajectories of meteors, space capsules are prime data gathering opportunities for the study of thermal protection materials at hypervelocity ( in this context, hypervelocity is defined as greater than escape velocity ).

Given a prime number that cannot be evenly divided into the total sample size.

Given any two reduced alternating diagrams D < sub > 1 </ sub > and D < sub > 2 </ sub > of an oriented, prime alternating link: D < sub > 1 </ sub > may be transformed to D < sub > 2 </ sub > by means of a sequence of certain simple moves called flypes.

Idealists are skeptics about the physical world, maintaining either: 1 ) that nothing exists outside the mind, or 2 ) that we would have no access to a mind-independent reality even if it may exist ; the latter case often takes the form of a denial of the idea that we can have unconceptualised experiences ( see Myth of the Given ).

Given the definition of above, we might fix ( or ' bind ') the first argument, producing a function of type.

Given a function f ∈ I < sub > x </ sub > ( a smooth function vanishing at x ) we can form the linear functional df < sub > x </ sub > as above.

Given the existence of a Godlike object in one world, proven above, we may conclude that there is a Godlike object in every possible world, as required.

Given our formula φ, we group strings of quantifiers of one kind together in blocks:

On poverty, Hoover said that " Given the chance to go forward with the policies of the last eight years, we shall soon with the help of God, be in sight of the day when poverty will be banished from this nation ", and promised, " We in America today are nearer to the final triumph over poverty than ever before in the history of any land ," but within months, the Stock Market Crash of 1929 occurred, and the world's economy spiraled downward into the Great Depression.

Given the state at some initial time ( t = 0 ), we can solve it to obtain the state at any subsequent time.

Given a complete set of axioms ( see below for one such set ), modus ponens is sufficient to prove all other argument forms in propositional logic, and so we may think of them as derivative.

Given a function ƒ defined over the reals x, and its derivative ƒ < nowiki > '</ nowiki >, we begin with a first guess x < sub > 0 </ sub > for a root of the function f. Provided the function is reasonably well-behaved a better approximation x < sub > 1 </ sub > is

Given that both A and not-A are seen to be “ true ,” Kant concludes that it ’ s not that “ God doesn ’ t exist ” but that there is something wrong with how we are asking questions about God and how we have been using our rational faculties to talk about universals ever since Plato got us started on this track!

Given how little we can know for sure, our focus should be on this earth and life ; beauty, justice, love.

Given any vector space V over K we can construct the tensor algebra T ( V ) of V. The tensor algebra is characterized by the fact:

Given metric spaces ( X, d < sub > 1 </ sub >) and ( Y, d < sub > 2 </ sub >), a function f: X → Y is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every x, y ∈ X with d < sub > 1 </ sub >( x, y ) < δ, we have that d < sub > 2 </ sub >( f ( x ), f ( y )) < ε.

Given the symmetry of circularly polarized light, we could have in fact selected any other two orthogonal components and found the same phase relationship between them.

Given an evaluation e of variables by elements of M < sub > w </ sub >, we

Given that any proposition containing conjunction, disjunction, and negation can be equivalently rephrased using conjunction and negation alone ( the conjunctive normal form ), we can now handle any compound proposition.

Given any energy eigenstate, we can act on it with the lowering operator, a, to produce another eigenstate with-less energy.

Given two ultrafilters and on, we define their sum by

Given a Boolean ring R, for x and y in R we can define

Given such possibilities, we can expect TC to be used to suppress everything from pornography to writings that criticize political leaders.

Given a testing procedure E applied to each prepared system, we obtain a sequence of values

Given objects and in an additive category, we can represent morphisms as-by-matrices

Given x ∈ A, the holomorphic functional calculus allows to define ƒ ( x ) ∈ A for any function ƒ holomorphic in a neighborhood of Furthermore, the spectral mapping theorem holds:

Given a preorder on S one may define an equivalence relation ~ on S such that a ~ b if and only if a b and b a.

Given a supervaluationist semantics, one can define the predicate ' supertrue ' as meaning " true on all precisifications ".

Given a differentiable manifold, one can unambiguously define the notion of tangent vectors and then directional derivatives.

Given some " primitive formulas " such as PM's primitives S < sub > 1 </ sub > V S < sub > 2 </ sub > OR, ~ S ( negation ) one is forced to define the axioms in terms of these primitive notions.

Given an inertial frame of reference and an arbitrary epoch ( a specified point in time ), exactly six parameters are necessary to unambiguously define an arbitrary and unperturbed orbit.

) Given a domain D we define a tuple over D as a partial function

Given a ring R and a two-sided ideal I in R, we may define an equivalence relation ~ on R as follows:

Given a subset V of A < sup > n </ sup >, we define I ( V ) to be the ideal of all functions vanishing on V:

Given a Riemannian manifold and two linearly independent tangent vectors at the same point, u and v, we can define

Given a manifold and a Lie algebra valued 1-form, over it, we can define a family of p-forms:

Given any associative superalgebra A one can define the supercommutator on homogeneous elements by

An open set U in X admits a local trivialization if and only if there exists a local section on U. Given a local trivialization one can define an associated local section by

Given an adapted process with define

Given a language L, and a pair of strings x and y, define a distinguishing extension to be a string z such that

Given a point x of a topological space X, and two maps f, g: X → Y ( where Y is any set ), then f and g define the same germ at x if there is a neighbourhood U of x such that restricted to U, f and g are equal ;

Given two column vectors and of random variables with finite second moments, one may define the cross-covariance to be the matrix whose entry is the covariance.

Given three points, if they are non-collinear, there are three pairs of parallel lines passing through them – choose two to define one line, and the third for the parallel line to pass through, by the parallel postulate.

Given a labelled state transition system (, Λ, →), define to be a function from binary relations over to binary relations over, as follows:

Given a flow network, and a flow on, we define the residual graph of with respect to as follows.

Given a function on, one may " geometrize " it by taking it to define a new manifold.

Given the image of a curve one can define several different parameterizations of the curve.