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* Every positive integer can be written as the sum of 73 or fewer sixth powers ( see Waring's problem ).

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## Some Related Sentences

Every and positive

__Every__non-zero number x, real

**or**complex, has n different complex number nth roots including any

__positive__

**or**negative roots,

**see**complex roots below.

__Every__line

**of**a GEDCOM file begins with a level number where all top-level records

**(**HEAD, TRLR, SUBN, and each INDI, FAM, OBJE, NOTE, REPO, SOUR, and SUBM ) begin with a line with level 0, while other level numbers are

__positive__integers.

__Every__

__positive__

**integer**

**can**

**be**expressed

**as**

**the**

**sum**

**of**at most 19 fourth

**powers**; every sufficiently large

**integer**

**can**

**be**expressed

**as**

**the**

**sum**

**of**at most 16 fourth

**powers**

**(**

**see**

**Waring's**

**problem**

**).**

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__C *- algebra has an approximate identity

**of**

__positive__elements

**of**norm ≤ 1 ; indeed,

**the**net

**of**all

__positive__elements

**of**norm ≤ 1 ; in A with its natural order always suffices.

'

__Every____positive__law,**or**every law simply and strictly so called, is set, directly**or**circuitously, by a sovereign person**or**body, to a member**or**members**of****the**independent political society wherein that person**or**body is supreme.__Every__residue class in this group contains exactly one square free

**integer**, and it is common, therefore, only to consider square free

__positive__integers, when speaking about congruent numbers.

Every and integer

*****

__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____integer__which**can****be****written****as****the****sum****of**two primes,**can**also**be****written****as****the****sum****of****as**many primes**as**one wishes, until all terms are units.*****

__Every__root

**of**a monic polynomial whose coefficients are algebraic integers is itself an algebraic

__integer__.

__Every__memory block in this system has an order, where

**the**order is an

__integer__ranging from 0 to a specified upper limit.

__Every__lattice in

**can**

**be**generated from a basis for

**the**vector space by forming all linear combinations with

__integer__coefficients.

In mathematics, an integer-valued polynomial

**(**also known**as**a numerical polynomial ) P**(**t ) is a polynomial whose value P**(**n ) is an__integer__for every__integer__n.__Every__polynomial with__integer__coefficients is integer-valued, but**the**converse is not true.*****

__Every__character value is a

**sum**

**of**n m < sup > th </ sup > roots

**of**unity, where n is

**the**degree

**(**that is,

**the**dimension

**of**

**the**associated vector space )

**of**

**the**representation with character χ and m is

**the**order

**of**g. In particular, when F is

**the**field

**of**complex numbers, every such character value is an algebraic

__integer__.

__Every__

__integer__instruction could operate on either 1-byte

**or**2-byte integers and could access data stored in registers, stored

**as**part

**of**

**the**instruction, stored in memory,

**or**stored in memory and pointed to by addresses in registers.

__Every__

__integer__2g ' 2g matrix with < sup >*</ sup > arises

**as**

**the**Seifert matrix

**of**a knot with genus g Seifert surface.

Every and can

__Every__such subset has a smallest element, so to specify our choice function we

__can__simply say that it maps each set to

**the**least element

**of**that set.

__Every__information exchange between living organisms — i. e. transmission

**of**signals that involve a living sender and receiver

__can__

**be**considered a form

**of**communication ; and even primitive creatures such

**as**corals are competent to communicate.

__Every__context-sensitive grammar which does not generate

**the**empty string

__can__

**be**transformed into an equivalent one in Kuroda normal form.

__Every__grammar in Chomsky normal form is context-free, and conversely, every context-free grammar

__can__

**be**transformed into an equivalent one which is in Chomsky normal form.

__Every__module over a division ring has a basis ; linear maps between finite-dimensional modules over a division ring

__can__

**be**described by matrices, and

**the**Gaussian elimination algorithm remains applicable.

__Every__entire function

__can__

**be**represented

**as**a power series that converges uniformly on compact sets.

Group actions / representations:

__Every__group G__can__**be**considered**as**a category with a single object whose morphisms are**the**elements**of**G. A functor from G to Set is then nothing but a group action**of**G on a particular set, i. e. a G-set.__Every__sequence

__can__, thus,

**be**read in three reading frames, each

**of**which will produce a different amino acid sequence

**(**in

**the**given example, Gly-Lys-Pro, Gly-Asn,

**or**Glu-Thr, respectively

**).**

__Every__hyperbola is congruent to

**the**origin-centered East-West opening hyperbola sharing its same eccentricity ε

**(**its shape,

**or**degree

**of**" spread "), and is also congruent to

**the**origin-centered North-South opening hyperbola with identical eccentricity ε — that is, it

__can__

**be**rotated so that it opens in

**the**desired direction and

__can__

**be**translated

**(**rigidly moved in

**the**plane ) so that it is centered at

**the**origin.

__Every__holomorphic function

__can__

**be**separated into its real and imaginary parts, and each

**of**these is a solution

**of**Laplace's equation on R < sup > 2 </ sup >.

__Every__species

__can__

**be**given a unique

**(**and, one hopes, stable ) name,

**as**compared with common names that are often neither unique nor consistent from place to place and language to language.

__Every__vector v in determines a linear map from R to taking 1 to v, which

__can__

**be**thought

**of**

**as**a Lie algebra homomorphism.

__Every__use

**of**modus tollens

__can__

**be**converted to a use

**of**modus ponens and one use

**of**transposition to

**the**premise which is a material implication.

__Every__adult, healthy, sane Muslim who has

**the**financial and physical capacity to travel to Mecca and

__can__make arrangements for

**the**care

**of**his / her dependants during

**the**trip, must perform

**the**Hajj once in a lifetime.

*****

__Every__finite topological space gives rise to a preorder on its points, in which x ≤ y if and only if x belongs to every neighborhood

**of**y, and every finite preorder

__can__

**be**formed

**as**

**the**specialization preorder

**of**a topological space in this way.

*****

__Every__preorder

__can__

**be**given a topology,

**the**Alexandrov topology ; and indeed, every preorder on a set is in one-to-one correspondence with an Alexandrov topology on that set.

__Every__binary relation R on a set S

__can__

**be**extended to a preorder on S by taking

**the**transitive closure and reflexive closure, R < sup >+=</ sup >.

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