For example, the prime number theorem states that the number of prime numbers less than or equal to N is asymptotically equal to N / ln N. Therefore the proportion of prime integers is roughly 1 / ln N, which tends to 0.

* For a finite field of prime order p, the algebraic closure is a countably infinite field which contains a copy of the field of order p < sup > n </ sup > for each positive integer n ( and is in fact the union of these copies ).

For example, the set with elements 2, 3, and 5 is equal to the set of all prime numbers less than 6.

For example, the Riemann hypothesis is a conjecture from number theory that ( amongst other things ) makes predictions about the distribution of prime numbers.

For 50 years, Coronation Street has remained at the centre of ITV's prime time schedule.

For example, primality testing is the problem of determining whether a given number is prime or not.

For each of the prime fields, one elliptic curve is recommended.

For example, neither 6 nor 35 is a prime number, since they both have two prime factors: 6 = 2 × 3 and 35 = 5 × 7.

The GCD of two numbers a and b is the product of the prime factors shared by the two numbers, where a same prime factor can be used multiple times, but only as long as the product of these factors divides both a and b. For example, since 1386 can be factored into 2 × 3 × 3 × 7 × 11, and 3213 can be factored into 3 × 3 × 3 × 7 × 17, the greatest common divisor of 1386 and 3213 equals 63 = 3 × 3 × 7, the product of their shared prime factors.

* For every prime number p and positive integer n, there exists a finite field with p < sup > n </ sup > elements.

For example, 9 and 28 are relatively prime.

For example, Campbell states that, " There have been systems of religion where the mother is the prime parent, the source ... We talk of Mother Earth.

For prime minister, he selected Odilon Barrot, an unobjectionable middle-road parliamentarian, who had led the " loyal opposition " under Louis-Philippe.

For most of Nehru's term as prime minister, India would continue to face serious food shortages despite progress and increases in agricultural production.

For instance, Fermat's little theorem for the nonzero integers modulo a prime generalizes to Euler's theorem for the invertible numbers modulo any nonzero integer, which generalizes to Lagrange's theorem for finite groups.

For instance, Goldbach's conjecture is the assertion that every even number ( greater than 2 ) is the sum of two prime numbers.

For example, a non-identity finite group is simple if and only if it is isomorphic to all of its non-identity homomorphic images, a finite group is perfect if and only if it has no normal subgroups of prime index, and a group is imperfect if and only if the derived subgroup is not supplemented by any proper normal subgroup.

For many people choosing an office suite, prime considerations include how valuable having learned that office suite will prove to potential employers, and how well the software interoperates with other users.

For example, 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2 and 3 in addition to 1 and 6.

For the aged horses: Mr. Budlong, p, 2:00.2, Lottie Thomas, p, 2:04.2, Mighty Signal 2:03, Clever Braden, p, 2:01.1, and Glow Star, p, 2:02.3 have been in 2:35 ; ;

For example, if K is a field of characteristic p and if X is transcendental over K, is a non-separable algebraic field extension.

For instance, the momentum operator p has the following form:

For F that are universal, such a p can generally be seen both as the concatenation of a program part and a data part, and as a single program for the function F.

For the re-reading by Benjamin Hudson, see Woolf, Pictland to Alba, pp. 127 – 129 and 152 – 157 ; Dumville, " Chronicle of the Kings of Alba ", p. 77.

For example, if the momentum p of an electron is said to be 1 GeV, then the conversion to MKS can be achieved by:

For closed ( orientable or non-orientable ) surfaces with positive genus, the maximum number p of colors needed depends on the surface's Euler characteristic χ according to the formula

For materials that absorb light, like metals and semiconductors, n is complex, and R < sub > p </ sub > does not generally go to zero.

For McNeely, Hermes is a god of the healing arts ( p. 88 ).

For Jung the trickster is the guide in total for the psychotherapeutic process ( p. 86 )

* Dan Woods, Peter Thoeny, Wikis for Dummies, For Dummies, 2007, ISBN 0-470-04399-7, p. 140

Thus Legendre's contribution lay in introducing a convenient notation that recorded quadratic residuosity of a mod p. For the sake of comparison, Gauss used the notation, according to whether a is a residue or a non-residue modulo p.

For typographical convenience, the Legendre symbol is sometimes written as ( a | p ) or ( a / p ).

For example, in the homonuclear diatomic molecules of the first ten elements, the molecular orbitals derived from the p < sub > x </ sub > and the p < sub > y </ sub > atomic orbitals result in two degenerate bonding orbitals ( of low energy ) and two degenerate antibonding orbitals ( of high energy ).

For example, the abelianized absolute Galois group G of is ( naturally isomorphic to ) an infinite product of the group of units of the p-adic integers taken over all prime numbers p, and the corresponding maximal abelian extension of the rationals is the field generated by all roots of unity.

For another example, see p-adic number.

For example, the rational numbers, the real numbers and the p-adic numbers have characteristic 0, while the finite field Z < sub > p </ sub > has characteristic p.

For example, this allows us to define F ( Z < sub > p </ sub >) with values in the p-adic numbers.

* For a fixed prime for any element different from zero write with such that does not divide, then is a valuation, called the p-adic valuation.

For example, over the p-adic numbers Hamilton's quaternions are a division algebra only when p is 2.

For odd prime p, the p-adic Hamilton quaternions are isomorphic to the 2 × 2 matrices over the p-adics.

For example, in accordance with the fashion of the times, most transitional societies have announced economic development plans of varying numbers of years ; ;

For one thing, although considerable numbers of men have been trained, bureaucracies are still deficient in many respects ; ;

For they considered the odd numbers as male and the even ones as female, equating the two groups with the Yang and Yin principles in Nature ; ;

For Euclid ’ s method to succeed, the starting lengths must satisfy two requirements: ( i ) the lengths must not be 0, AND ( ii ) the subtraction must be “ proper ”, a test must guarantee that the smaller of the two numbers is subtracted from the larger ( alternately, the two can be equal so their subtraction yields 0 ).

For example, suppose that each member of the collection X is a nonempty subset of the natural numbers.

For example, suppose that X is the set of all non-empty subsets of the real numbers.

For example, while the axiom of choice implies that there is a well-ordering of the real numbers, there are models of set theory with the axiom of choice in which no well-ordering of the reals is definable.

For example, if we abbreviate by BP the claim that every set of real numbers has the property of Baire, then BP is stronger than ¬ AC, which asserts the nonexistence of any choice function on perhaps only a single set of nonempty sets.

For example an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces.

For, the letters associated with those numbers are K, L, M, N, O, ..., respectively.

For example, the number of solutions of an equation over a finite field reflects the topological nature of its solutions over the complex numbers.

For socket numbers, a few well-known numbers were reserved for special purposes specific to the AppleTalk protocol itself.

For example, the field extension R / Q, that is the field of real numbers as an extension of the field of rational numbers, is transcendental, while the field extensions C / R and Q (√ 2 )/ Q are algebraic, where C is the field of complex numbers.

For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers.

For example, if the numbers involved in a computation may be arbitrarily large, the time required by a single addition can no longer be assumed to be constant.

For this reason, parallel computing is only useful for either small numbers of processors, or problems with very high values of P: so-called embarrassingly parallel problems.

For example, the smallest pair of amicable numbers is ( 220, 284 ); for the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284 ; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which the sum is 220.

For small numbers of at-bats, it is possible ( though unlikely ) for a player's on-base percentage to be lower than his batting average ( H / AB ).

For example, the division example above may also be interpreted as a partial binary function from Z and N to Q, where N is the set of all natural numbers, including zero.

For instance, division of real numbers is a partial function, because one can't divide by zero: a / 0 is not defined for any real a.