 Page "Fundamental theorem of arithmetic" ¶ 4
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Every and positive Every positive number a has two square roots:, which is positive, and, which is negative. < li > Every positive definite matrix is invertible and its inverse is also positive definite. Every positive integer appears exactly once somewhere on this list. Every positive real number x has a single positive nth root, which is written. 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 rational number can be represented by an Egyptian fraction. * Every subset of may be covered by a finite set of positive orthants, whose apexes all belong to * Every positive integer can be written as the sum of 73 or fewer sixth powers ( see Waring's problem ). Every positive integer is the sum of at most 37 fifth powers ( see Waring's problem ). * Every positive integer except 1 is a PV number. 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 positive rational number can be expanded as an Egyptian fraction. Every positive integer is the sum of at most 143 seventh powers ( see Waring's problem ). 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 positive rational number q may be expressed as a continued fraction of the form

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 real number, whether integer, rational, or irrational, has a unique location on the line. : Every even integer greater than 2 can be expressed as the sum of two primes. : 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 integer greater than 2 can be written as the sum of three primes. : Every integer greater than 5 can be written as the sum of three primes. : Every even integer greater than 2 can be written as the sum of two primes, * 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 integer greater than 96 may be represented as a sum of distinct super-prime numbers. 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 polynomial ring R ..., x < sub > n </ sub > is a commutative R-algebra. * 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 ketose will have 2 < sup >( n-3 )</ sup > stereoisomers where n > 2 is the number of carbons. Every aldose will have 2 < sup >( n-2 )</ sup > stereoisomers where n > 2 is the number of carbons. 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 finite tree with n vertices, with, has at least two terminal vertices ( leaves ). 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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