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 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.

Given its exposed spot on the top of Stow Hill, the town is often referred to with the couplet " Stow on the Wold, where the winds blow cold and the cooks can't roast their dinners ", but there is no source for this.

Given good viewing conditions, Thuban is relatively easy to spot in the night sky, due to its location in relation to the Big Dipper asterism of Ursa Major.

Dalglish brought in Shay Given, costing £ 1. 5 million, and a season-long battle for the number one spot ensued between the two goalkeepers.

Given a high-profile spot on the all-star compilation Roll Wit tha Flava as their first recording opportunity, Zhane lived up to the pressure and came away with one of the hip-hop party anthems of all time, " Hey, Mr. D. J.

Given the oversight authority granted to DCAA, auditors are trained to spot access to records violations.

Given strong evidence that CIRP holds, the forward rate unbiasedness hypothesis can serve as a test to determine whether UIRP holds ( in order for the forward rate and spot rate to be equal, both CIRP and UIRP conditions must hold ).

Given the similarity of their records, it was likely the outcome would have playoff implications, since the first tie-breaker for a wild card spot is the head-to-head record.

: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.

Given this, the bidding is said to start at the one-level when contracting for a total of seven tricks, at the two-level for eight tricks and so on to the seven-level to contract to take all thirteen tricks.

Given a surface through which a metal wire passes, electrons move in both directions across the surface at an equal rate.

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 age of the Rig Veda being at least 1500 BC or before, this must rank as the earliest practice of beating of the war drum in the history of mankind.

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 the limited budget for schools, it is claimed that a voucher system would weaken public schools while at the same time not necessarily providing enough money for people to attend private schools.

* Frederick Douglass lecture on Haiti – Given at the World's Fair in Chicago, January 1893.

Given away at theatrical showings was a comic herald that told a simplified version of the film.

Given a complex-valued function ƒ of a single complex variable, the derivative of ƒ at a point z < sub > 0 </ sub > in its domain is defined by the limit

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

Along with a hand-picked escort led by Given Campbell, Davis and his wife were captured on May 10, 1865, at Irwinville in Irwin County, Georgia.

Given civilian status, he recovered his teaching position at Lycée Pasteur near Paris, settled at the Hotel Misgiven a new position at Lycée Condorcet, replacing a Jewish teacher who had been forbidden to teach by Vichy law.

Given that bodhisattvas are known to incarnate at will as living people according to the sutras, the princess Miaoshan is generally viewed as an incarnation of Avalokitesvara.

Given the choice between nylon and gut, many luthiers prefer to use gut, as it conforms more readily to the sharp angle at the edge of the fingerboard.

Given the need for such fine regulation of Ca < sup > 2 +</ sup > signaling, it is perhaps unsurprising that dysregulated mitochondrial Ca < sup > 2 +</ sup > has been implicated in several neurodegenerative diseases, while the catalogue of tumor suppressors includes a few that are enriched at the MAM.

Given printing practices at the time ( which included type-setting from dictation ), no two editions turned out to be identical, and it is relatively rare to find even two copies that are exactly the same.

Given a year to complete the work, Gibson undertook the actual writing out of " blind animal terror " at the obligation to write an entire novel – a feat which he felt he was " four or five years away from ".

Given the task of overhauling the Cardinals by owner Gussie Busch, Herzog was looking to trade Templeton when he was approached by Padres General Manager Jack McKeon at the 1981 baseball winter meetings.

Given prevailing views on homosexuality at the time, circulation of the diaries helped undermine support for clemency for Casement.

Given their wide historical distribution, red wolves probably utilized a large suite of habitat types at one time.