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* For every prime number p and positive integer n, there exists a finite field with p < sup > n </ sup > elements.
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For and every
For every criterion which defines what something is, at the same time proclaims -- implicitly if not openly -- what that something is not.
For new galaxies to be created, Professor Bondi declares, it would only be necessary for a single hydrogen atom to be created in an area the size of your living room once every few million years.
For example, " biweekly " can mean " fortnightly " ( once every two weeks – 26 times a year ), or " twice a week " ( 104 times a year ).
** Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A × A.
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 Tarrou, plague is the destructive impulse within every person, the will and the capacity to do harm, and it is everyone's duty to be on guard against this tendency within themselves, lest they infect someone else with it.
For every group G there is a natural group homomorphism G → Aut ( G ) whose image is the group Inn ( G ) of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can be embedded into its own automorphism group.
For Hume, every effect only follows its cause arbitrarily — they are entirely distinct from one another.
Professor Henry Higgins sings, " Look at her, a prisoner of the gutters / Condemned by every syllable she utters / By right she should be taken out and hung / For the cold-blooded murder of the English tongue.
For example, the division example above is surjective ( or onto ) because every rational number may be expressed as a quotient of an integer and a natural number.
For example, the BBC website, which had previously been called BBC Online, took on the BBCi brand from 2001, displaying an i-bar across the top of every page, offering a category-based navigation: Categories, TV, Radio, Communicate, Where I Live, A-Z Index, and a search.
For example, in the Schrödinger picture, there is a linear operator U with the property that if an electron is in state right now, then in one minute it will be in the state, the same U for every possible.
For and prime
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 example, primality testing is the problem of determining whether a given number is prime or not.
For a prime p, the p-adic numbers arise by completing the rational numbers with respect to a different metric.
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 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.
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