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 prime, we define the height of, written to be the supremum of the set

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 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 algebraic number, there is a unique monic polynomial ( with rational coefficients ) of least degree that has the number as a root.

Given the large number of U. S. military personnel and their dependents residing in Europe, it was expected that over 7 % of donors would be deferred due to the policy.

Given that a natural language such as English contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or rely upon primitive notions.

Given is each element's name, atomic number, year of first report, name of the discoverer, and some notes related to the discovery.

The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.

Given infinite space, there would, in fact, be an infinite number of Hubble volumes identical to ours in the universe.

Given the increasing number of scientists who study the nervous system, several prominent neuroscience organizations have been formed to provide a forum to all neuroscientists and educators.

: Given any positive number ε, there is a sequence

Given the number of homes, it seems likely that no more than fifty people lived in Skara Brae at any given time.

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 that detecting a price cut becomes harder as the number of firms increases, the bigger are the gains from price cutting.

Given the ( period, cash flow ) pairs (, ) where is the total number of periods, the net present value is given by:

* Aggregate signature-a signature scheme that supports aggregation: Given n signatures on n messages from n users, it is possible to aggregate all these signatures into a single signature whose size is constant in the number of users.

Given the ( period, cash flow ) pairs (, ) where is a positive integer, the total number of periods, and the net present value, the internal rate of return is given by in:

( Please note that Japan has a guideline of a limit of 1 % of GDP on defense spending, however, Japan defines a number of activities as non-defense spending ) Given these circumstances, some have viewed Article 9 as increasingly irrelevant.

Given the rules of any two-person game with a finite number of positions, one can always trivially construct a minimax algorithm that would exhaustively traverse the game tree.

Given that the Masoretes would not alter the sacred consonantal text, the Kethiv-Qere notes were a way of " correcting " or commenting on the text for any number of reasons ( grammatical, theological, aesthetic, etc.

Given the number of unpleasant, harmful or even fatal consequences of coming out in world societies, it is questionable to call being closeted a choice.

Given any input, GEN generates an infinite number of candidates, or possible realizations of that input.

Given the number of times tea is poured in a meal, the tapping is a timesaver in loud restaurants or lively company, as an individual being served might be speaking to someone else or have food in their mouth.

Given an input consisting of any number of bytes, it produces as output a single byte that is strongly dependent on every byte of the input.

Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.

Given the fact that the Lakeport area has an economy that is based somewhat on agricultural activities occurring in the Big Valley and Scotts Valley area which involve the growing and harvesting of food and nut crops as well as the increasing wine industry, there are a significant number of transient and seasonal farm workers.

Given that one cannot be motivated by reason alone, requiring the input of the passions, Hume argued that reason cannot be behind morality.

Given that white is the most effective color in terms of facilitating the reflection of ground light, the hypothesis that straight hair could have been adaptively favorable, cannot be fully discounted in this regard.

Given the same set of verifiable facts, some societies or individuals will have a fundamental disagreement about what one ought to do based on societal or individual norms, and one cannot adjudicate these using some independent standard of evaluation.

Given the proposition that ' a persons life prospects should not be decided by factors outside of their control or for which a person cannot claim personal credit ' ( i. e. social status, inherited wealth, race and other accidents of birth ) a meritocracy proposes a system where people are rewarded based on their efforts, and if everyone can start on equal footing with the same opportunity to advance, then the results are just.

Given these assumptions, the constraint that time travel must not lead to inconsistent outcomes could be seen merely as a tautology, a self-evident truth that cannot possibly be false, because if you make the assumption that it is false this would lead to a logical paradox.

The wavelets forming a continuous wavelet transform ( CWT ) are subject to the uncertainty principle of Fourier analysis respective sampling theory: Given a signal with some event in it, one cannot assign simultaneously an exact time and frequency response scale to that event.

: Given that desegregation has not produced the predicted leaps forward in black educational achievement, there is no reason to think that black students cannot learn as well when surrounded by members of their own race as when they are in an integrated environment.

Given his role in the usurpation of Benedict IX, and his attitude towards Henry III, it is unsurprising that Boniface at first refused, advising Poppo when he entered Tuscany, " I cannot go to Rome with you.

When the measure was sent in May to the parliament they said " Given the damage it produces on those rules which allow the life in community, ensure the dignity of the person and equality between sexes, this practice, even if it is voluntary, cannot be tolerated in any public place ".

Given that humanity cannot exist except within a covenantal relationship with God, and all covenants use symbols to give us " the attestation of his grace ", he gives the tree, " not because it could confer on man that life with which he had been previously endued, but in order that it might be a symbol and memorial of the life which he had received from God.

Given the morphological distinctness of the Cape Verde birds and the fact that the Cape Verde population was isolated from other populations of Red Kites, it cannot be conclusively resolved at this time whether the Cape Verde population was not a distinct subspecies ( as M. migrans fasciicauda ) or even species that frequently absorbed stragglers from the migrating European populations into its gene pool.

Heterogeneity / Variability: Given the very nature of services, each service offering is unique and cannot be exactly repeated even by the same service provider.

* Given only one list item, one cannot immediately obtain the addresses of the other elements of the list.

Given the context in which it is used, as well as the evidence of Swift's writing, the possibility that " bloody " is also a minced oath ( or more precisely, a slang usage of an otherwise legitimate word masquerading as a minced oath ) cannot be lightly dismissed.

Given the lack of surviving objects, we cannot know how common the techniques employed were, but the quality of the execution suggests that the binder was experienced in them.

Given this and the personal animosity towards Dowding the Air Ministry view of the Battle of Britain cannot be said to be " even-handed ".

Given that we have been on " yellow alert " for over two and a half years now, we cannot consider this a particularly exceptional condition that warrants curtailment of constitutional rights.

Given the lack of regulatory standards in the U. S., some public-health researchers have raised the concern that consumers cannot be certain that spirulina and other blue-green algae supplements are free of contamination.

Given that Jack the Ripper actually did show up on the station last year, the possibility that he is telling the truth cannot be dismissed out of hand.

Given that there is only one binary operation, distributivity obviously cannot be defined in the standard way.

The SPR went on to say the stories relating to battlefield " visions " which circulated during the spring and summer of 1915, " prove on investigation to be founded on mere rumour, and cannot be traced to any authoritative source ".” Given that the Society for Psychical Research believed in the existence of supernatural forces, the conclusions of this report are highly significant.

Given the example above, the two galaxies in question cannot have shared any sort of information ; they are not in " causal contact ".

Given the fact that many working poor people do not own a car or cannot afford to drive their car, where they live can significantly limit where they are able to work, and vice versa.

Given that many governments cannot be legally bound by a non-governmental document such as the World Anti-Doping Code, they are implementing it by individually ratifying the UNESCO International Convention against Doping in Sport, the first global international treaty against doping in sport, which was unanimously adopted by 191 governments at the UNESCO General Conference in October 2005 and came into force in February 2007.