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Every and prime
Every positive integer n > 1 can be represented in exactly one way as a product of prime powers:
Hilbert's example: " the assertion that either there are only finitely many prime numbers or there are infinitely many " ( quoted in Davis 2000: 97 ); and Brouwer's: " Every mathematical species is either finite or infinite.
Every completely multiplicative function is a homomorphism of monoids and is completely determined by its restriction to the prime numbers.
* In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i. e. M is contained in exactly 2 ideals of R, namely M itself and the entire ring R. Every maximal ideal is in fact prime.
# Every prime ideal of A is principal.
Every third odd number is divisible by 3, which requires that no three successive odd numbers can be prime unless one of them is 3.
* Hardy and Littlewood listed as their Conjecture I: " Every large odd number ( n > 5 ) is the sum of a prime and the double of a prime.
Every prime ideal P in a Boolean ring R is maximal: the quotient ring R / P is an integral domain and also a Boolean ring, so it is isomorphic to the field F < sub > 2 </ sub >, which shows the maximality of P. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings.
Every strong probable prime to base a is also an Euler probable prime to the same base, but not vice versa.
Every closed 3-manifold has a prime decomposition: this means it is the connected sum of prime three-manifolds ( this decomposition is essentially unique except for a small problem in the case of non-orientable manifolds ).
: Every oriented prime closed 3-manifold can be cut along tori, so that the interior of each of the resulting manifolds has a geometric structure with finite volume.
Every non-zero characteristic is a prime number.
: Every Boolean algebra contains a prime ideal.
* Every number of the form 2 < sup > m </ sup > p for a natural number m and a prime number p such that p < 2 < sup > m + 1 </ sup > is also semiperfect.
Every group of prime order is cyclic, since Lagrange's theorem implies that the cyclic subgroup generated by
Iwasawa worked with so-called-extensions: infinite extensions of a number field with Galois group isomorphic to the additive group of p-adic integers for some prime p. Every closed subgroup of is of the form, so by Galois theory, a-extension is the same thing as a tower of fields such that.
Every imperfect field is necessarily transcendental over its prime subfield ( the minimal subfield ), because the latter is perfect.
A finitely-generated abelian group is indecomposable if and only if it is isomorphic to Z or to a factor group of the form for some prime number p and some positive integer n. Every finitely-generated abelian group is a direct sum of ( finitely many ) indecomposable abelian groups.
* Every non-split, prime, alternating link that is not a torus link is hyperbolic by a result of William Menasco.

Every and pair
Every ATM cell has an 8-or 12-bit Virtual Path Identifier ( VPI ) and 16-bit Virtual Channel Identifier ( VCI ) pair defined in its header.
* Every pair of congruence relations for an unknown integer x, of the form x ≡ k ( mod a ) and x ≡ l ( mod b ), has a solution, as stated by the Chinese remainder theorem ; in fact the solutions are described by a single congruence relation modulo ab.
Every pair of points belongs to a unique line.
Every Saturday night, from 1995 to 2000, he would host a pair of horror films ( such as Friday the 13th Part 2 and Wes Craven's New Nightmare ) provide a running commentary, trivia, off-color jokes and a drive-in total.
This pair of brothers later wrote a book entitled Foot by Foot Through the USA, A High Adventure Odyssey To Every State in the Union, which chronicled their journey.
Every constraint is in turn a pair ( usually represented as a matrix ), where is an-tuple of variables and is an-ary relation on.
Every minimal pair of natural numbers has and, for otherwise the pair would belong to and be at least as small in both coordinates, contradicting the assumption that is minimal.
Every pair of MojoNation nodes maintained a relative credit balance, with every EGTP request transferring some Mojo credit from the sender to the receiver.
Every pair of colors appears on at least one edge.
Every Wednesday the pair write about everything from summer holidays to sexy ladies.
Every point in the training is an input-output pair, where the input
Every pair plays against a number of opposing pairs in successive rounds, depending on the size of the field.
Every homeroom is led by a pair of teachers with combined native language abilities in both English and Mandarin, and at every year level, teachers of the two languages collaborate to plan and teach outcomes from the school's inquiry-based integrated program.
* Every pair of distinct points determines a unique line.
Every matched pair of threads, external and internal, can be described as male and female.
Knopfler performed " Money for Nothing " using his Pensa-Suhr signature MK-1 model guitar with a pair of Soldano SLO-100 tube / valve amplifier heads and Marshall speaker cabinets during the Nelson Mandela 70th Birthday Tribute and Prince's Trust concerts in 1988, as well as the Nordoff-Robbins charity show at Knebworth in 1990 and the On Every Street world tours in 1991 / 1992.
Every pair of balls ( Hamming spheres ) are non-intersecting by the t-error-correcting property, and each ball contains ( in other words, the volume of the ball ) m words.
As described in an article in the London newspaper, The Daily Courier, on June 25, 1896: " Every one who has watched a bout with the foils knows that the task of judging the hits is with a pair of amateurs difficult enough, and with a well-matched pair of maîtres d ’ escrime well-nigh impossible.
Every pair of points are i < sup > th </ sup > associates for exactly one.
Every convex centrally symmetric polyhedron P in R < sup > 3 </ sup > admits a pair of opposite ( antipodal ) points and a path of length L joining them and lying on the boundary ∂ P of P, satisfying
Every complex number has both a real part and an imaginary part, so one complex variable is two-dimensional and a pair of complex variables is four-dimensional.

Every and except
Every member of the future protectionist cabinet then in parliament ( except Disraeli ) voted against the measure.
Every action specifically intended to prevent procreation is forbidden, except in medically necessary circumstances.
Every president since Taft, except for Jimmy Carter, threw out at least one ceremonial first ball or pitch for Opening Day, the All-Star Game, or the World Series, usually with much fanfare.
Every S / 360 ( except for the Model 20, which was not a standard S / 360 ) had a byte-multiplexer channel and 1 or more selector channels.
Every element s, except a possible greatest element, has a unique successor ( next element ), namely the least element of the subset of all elements greater than s. Every subset which has an upper bound has a least upper bound.
Every album, except 1969's The Chicago Transit Authority, 1978's Hot Streets and 1995's Night and Day: Big-Band, released by Chicago Transit Authority / Chicago has been the band name followed by a Roman numeral or numbered in some other manner.
Every state except Nebraska has a bicameral legislature, meaning that the legislature consists of two separate legislative chambers or houses.
Every deity and demi-god had been invited, except Eris, the goddess of strife ( no one wanted a troublemaker at a wedding ).
Every book in the Hebrew Bible was represented in this discovery except Esther.
Every corn-growing country decided to increase tariffs in reaction to this, except Britain and Belgium.
Every British North Atlantic route was suspended until 1856 except Cunard's Liverpool-Halifax-Boston service.
Every evening since 1928 ( except for a period during the Second World War when Ypres was occupied by Germany ), at precisely eight o ' clock, traffic around the imposing arches of the Menin Gate Memorial has been stopped while the Last Post is sounded beneath the Gate by the local fire brigade.
Every body panel, except for the doors, were redesigned.
Every sequence in principle has a generating function of each type ( except that Lambert and Dirichlet series require indices to start at 1 rather than 0 ), but the ease with which they can be handled may differ considerably.
Every day had the name of an agricultural plant, except the 5th ( Quintidi ) and 10th day ( Decadi ) of every decade, which had the name of a domestic animal ( Quintidi ) or an agricultural tool ( Decadi ).
Every episode except one featured a live performance by a band, including Madness, Motörhead, and The Damned.
Every century except the 21st has seen major building work or adaptations at the castle.
Every state, except Arkansas and Mississippi, maintains a state FEPA ( see EEOC and state FEPA directory ).
* Every real number greater than zero or every complex number except 0 has two square roots.
Every day had the name of an agricultural plant, except the 5th ( Quintidi ) and 10th day ( Decadi ) of every decade, which had the name of a domestic animal or an agricultural tool, respectively.
Every song features Carly Simon singing except for ' The More It Snows ( Tiddely-Pom )'.
Every province in Canada except Alberta has either a Provincial Sales Tax ( PST ) or the Harmonized Sales Tax ( HST ), which is a single, blended combination of the GST and PST.

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