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Every positive integer n > 1 can be represented in exactly one way as a product of prime powers:

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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**written**as**the sum**of**73 or fewer sixth**powers**( see Waring's problem ).__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 n

*

__Every__quadratic Bézier curve is also**a**cubic Bézier curve, and more generally, every degree__n__Bézier curve is also**a**degree m curve for any m**>**__n__. In detail,**a**degree__n__curve with control points P < sub**>**0 </ sub >, …, P < sub**>**__n__</ sub**>**is equivalent ( including the parametrization ) to the degree__n__+**1**curve with control points P '< sub**>**0 </ sub >, …, P '< sub**>**__n__+**1**</ sub >, where.__Every__twin

**prime**pair except ( 3, 5 ) is

**of**the form ( 6n −

**1**, 6n +

**1**) for some natural number

__n__, and with the exception

**of**< var

**>**

__n__</ var

**>**=

**1**, < var

**>**

__n__</ var

**>**must end

**in**0, 2, 3, 5, 7, or 8.

* 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__random vector gives rise to

**a**probability measure on R < sup

**>**

__n__</ sup

**>**with the Borel algebra

**as**the underlying sigma-algebra.

*

__Every__left ideal I**in**R is finitely generated, i. e. there exist elements**a**< sub**>****1**</ sub >, ...,**a**< sub**>**__n__</ sub**>****in**I such that I = Ra < sub**>****1**</ sub**>**+ ... + Ra < sub**>**__n__</ sub >.__Every__smooth submanifold

**of**R < sup

**>**

__n__</ sup

**>**has an induced Riemannian metric g

**:**the inner

**product**on each tangent space is the restriction

**of**the inner

**product**on R < sup

**>**

__n__</ sup >.

__Every__finite group

**of**exponent

__n__with m generators is

**a**homomorphic image

**of**B < sub

**>**0 </ sub >( m,

__n__).

__Every__smooth manifold defined

**in**this

**way**has

**a**natural diffeology, for which the plots correspond to the smooth maps from open subsets

**of**R < sup

**>**

__n__</ sup

**>**to the manifold.

__Every__known Sierpinski number k has

**a**small covering set,

**a**finite set

**of**primes with at least

**one**dividing k · 2 < sup

**>**

__n__</ sup >+

**1**for each

__n__

**>**0.

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