Copyright © Philip M. Parker, INSEAD. Terms of Use.
Definition: Prime |
PrimeAdjective1. First in rank or degree; "an architect of premier rank"; "the prime minister". 2. Used of the first or originating agent; "prime mover". 3. Of superior grade; "choice wines"; "prime beef"; "prize carnations"; "quality paper"; "select peaches". 4. (math) of or relating to or being an integer that cannot be factored into other integers; "prime number". 5. At the best stage; "our manhood's prime vigor"- Robert Browning. Noun1. A number that has no factor but itself and 1. 2. The period of greatest prosperity or productivity. 3. The second canonical hour; about 6 a.m. 4. The time of maturity when power and vigor are greatest. Verb1. Insert a primer into (a gun, mine, charge, etc.) preparatory to detonation or firing; "prime a cannon"; "prime a mine". 2. Cover with a primer; apply a primer to. 3. Fill with priming liquid; "prime a car engine". Source: WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved. |
Date "prime" was first used in popular English literature: sometime before 1321. (references) |
Note: Prime \Prime\, transitive verb. [imperfect & past participle. Primed; Priming.]. (references) |
| Domain | Definition |
Chemical Industry | To inject fuel directly into the cylinders of an internal combbustion engine, to make starting easier. Source: European Union. (references) |
Food & Agriculture | The highest grade of timber, etc. in various grading rules. Source: European Union. (references) |
Literature | Prime (l syl.). In the Catholic Church the first canonical hour after lauds. Milton terms sunrise "that sweet hour of prime." (Paradise Lost, bk. v. 170.) "All night long ... came the sound of chanting ... as the monks sang the service of matins, lauds, and prime."- Shorthouse: John Inglesant, chap. 1. p. 10. Source: Brewer's Dictionary. |
Mathematics | M is not divisible by the square of any --. Source: European Union. (references) |
Metallurgy | To cover(a surface)with a preparatory coat or colour, before painting it. Source: European Union. (references) |
Slang in 1811 | PRIME. Bang up. Quite the thing. Excellent. Well done. She's a prime piece; she is very skilful in the venereal act. Prime post. She's a prime article. Source: 1811 Dictionary of the Vulgar Tongue. |
Sports & Leisure | Hand pronated, point of blade below bell, protects inside. . low. . /. Source: European Union. (references) |
Source: compiled by the editor from various references; see credits. | |
(From Wikipedia, the free Encyclopedia)
In abstract algebra, an integral domain, is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero. Integral domains are generalizations of the integers and provide a natural setting for studying divisibility.Alternatively and equivalently, integral domains may be defined as commutative rings in which the zero ideal {0} is prime, or as the subrings of fields.
Examples
The prototypical example is the ring Z of all integers.
Every field is an integral domain. Conversely, every Artinian integral domain is a field. In particular, the only finite integral domains are the finite fields.
Rings of polynomials are integral domains if the coefficients come from an integral domain. For instance, the ring Z[X] of all polynomials in one variable with integer coefficients is an integral domain; so is the ring R[X,Y] of all polynomials in two variables with real coefficients .
The set of all real numbers of the form a + b√2 with a and b integers is a subring of R and hence an integral domain. A similar example is given by the complex numbers of the form a + bi with a and b integers (the Gaussian integers).
If U is a connected open subset of the complex number plane C, then the ring H(U) consisting of all holomorphic functions f : U -> C is an integral domain. The same is true for rings of analytical functions on connected open subsets of analytical manifolds.
If R is a commutative ring and P is a prime ideal in R, then the factor ring R/P is an integral domain.
Divisibility, prime and irreducible elements
If a and b are elements of the integral domain R, we say that a divides b or a is a divisor of b or b is a multiple of a if and only if there exists an element x in R such that ax = b.
If a divides b and b divides c, then a divides c. If a divides b, then a divides every multiple of b. If a divides two elements, then a also divides their sum and difference.
The elements which divide 1 are called the units of R; these are precisely the invertible elements in R. Units divide all other elements.
If a divides b and b divides a, then we say a and b are associated elements. a and b are associated if and only if there exists a unit u such that au = b.
If q is a non-unit, we say that q is an irreducible element if q cannot be written as a product of two non-units.
If p is a non-zero non-unit, we say that p is a prime element if, whenever p divides a product ab, then p divides either a or b.
This generalizes the ordinary definition of prime number in the ring Z, except that it allows for negative prime elements. If p is a prime element, then the principal ideal (p) generated by p is a prime ideal. Every prime element is irreducible (here, for the first time, we need R to be an integral domain), but the converse is not true in all integral domains (it is true in unique factorization domains, however).
Field of fractions
If R is a given integral domain, the smallest field Quot(R) containing R as a subring is uniquely determined up to isomorphism and is called the field of fractions or quotient field of R. It consists of all fractions a/b with a and b in R and b ≠ 0. The field of fractions of the integers is the field of rational numbers. The field of fractions of a field is that field itself.
Characteristic and homomorphisms
The characteristic of every integral domain is either zero or a prime number.
If R is an integral domain with prime characteristic p, then f(x) = xp defines an injective ring homomorphism f : R -> R, the Frobenius homomorphism.
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Integral domain."
(From Wikipedia, the free Encyclopedia)
Prime can refer to:
- Prime number
- Prime (ribose structure)
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Prime."
(From Wikipedia, the free Encyclopedia)
In abstract algebra, the concept of prime ideals is an important generalization of the concept of prime numbers. If R is a commutative ring, then an ideal P of R is called prime if it has the following two properties:This generalizes the following property of prime numbers: if p is a prime number and if p divides a product ab of two integers, then p divides a or p divides b. We can therefore say
- whenever a, b are two elements of R such that their product ab lies in P, then a is in P or b is in P.
- P is not equal to the whole ring R
- A positive integer n is a prime number if and only if the ideal Zn is a prime ideal in Z.
Examples
- If R denotes the ring C[X, Y] of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial Y2 - X3 - X - 1 is a prime ideal (see elliptic curve).
- In the ring Z[X] of all polynomials with integer coefficients, the ideal generated by 2 and X is a prime ideal. It consists of all those polynomials whose constant coefficient is even.
- In any ring R, a maximal ideal is an ideal M that is a subset of exactly 2 ideals (which must then be M itself and the entire ring R). Every maximal ideal is in fact prime.
- If M is a smooth manifold, R is the ring of smooth functions on M, and x is a point in M, then the set of all smooth functions f with f(x) = 0 forms a prime ideal (even a maximal ideal) in R.
Properties
- An ideal I in the commutative ring R is prime if and only if the factor ring R/I is an integral domain.
- Every maximal ideal (see above) is prime; an ideal I in the commutative ring R is a maximal ideal if and only if the factor ring R/I is a field.
- Every commutative ring ≠ 0 contains at least one prime ideal. In fact, it contains at least one maximal ideal, which can be proven using Zorn's lemma.
- A commutative ring is an integral domain if and only if {0} is a prime ideal.
- A commutative ring is a field if and only if {0} is its only prime ideal, or alternatively, if and only if {0} is a maximal ideal.
Uses
One use of prime ideals occurs in algebraic geometry, where varieties are defined as the zero sets of ideals in polynomial rings. It turns out that the irreducible varieties correspond to prime ideals. In the modern abstract approach, one starts with an arbitrary commutative ring and turns the set of its prime ideals, also called its spectrum, into a topological space and can thus define generalizations of varieties called schemes, which find applications not only in geometry, but also in number theory.
The introduction of prime ideals in algebraic number theory was a major step forward: it was realized that the ordinary fundamental theorem of arithmetic does not work in rings of algebraic integers, but a substitute was found when Dedekind replaced elements by ideals and prime elements by prime ideals; see Dedekind domain.
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Prime ideal."
(From Wikipedia, the free Encyclopedia)
In mathematics, a prime number, or prime for short, is a natural number larger than 1 that has as its only positive divisors (factors) 1 and itself. The first 25 prime numbers are
This definition is used throughout the Wikipedia. See the Generalizations section, below, for another definition in common use. The property of being a prime is called primality. If a number greater than one is not a prime number it is called a composite number.
- 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
Representing natural numbers as products of primes
An important fact is the fundamental theorem of arithmetic, which states that every natural number can be written as a product of primes, and in essentially only one way. Primes are thus the "basic building blocks" of the natural numbers. For example, we can write
See Prime factorization algorithm for details.
- .
How many prime numbers are there?
There are infinitely many prime numbers. The oldest known proof for this statement is given by the Greek mathematician Euclid in his Elements (Book IX, Proposition 20). Euclid states the result as "there are more than any given [finite] number of primes", and his proof can be briefly summarized as follows:
Other mathematicians have given their own proofs. One of those (due to Euler) shows that the sum of the reciprocals of all prime numbers diverges to infinity. Kummer's is particularly elegant and Furstenberg provides one using topological terms.
- Take a finite number of primes. Multiply them all together and add one. The resulting number is not divisible by any of the finite set of primes, because dividing by any of these would give a remainder of one. Therefore it must be divisible by some prime that was not included in the finite set.
Even though the total number of primes is infinite, one could still ask "how many primes are there below 100,000" or "How likely is a random 100-digit number to be prime?" Questions like these are answered by the prime number theorem.
Finding prime numbers
The Sieve of Eratosthenes is a simple way to compute the list of all prime numbers up to a given limit.
In practice though, one usually wants to check if a given number is prime, rather than generate a list of primes. Further, it is often satisfactory to know the answer with a high probability. It is possible to quickly check whether a given large number (say, up to a few thousand digits) is prime using probabilistic primality tests. These typically pick a random number called a "witness" and check some formula involving the witness and the potential prime N. After several iterations, they declare N to be "definitely composite" or "probably prime". These tests are not perfect. For a given test, there may be some composite numbers that will be declared "probably prime" no matter what witness is chosen. Such numbers are called pseudoprimes for that test. Here's a description of the Fermat primality test.
A new algorithm which determines whether a given number N is prime and which uses time polynomial in the number of digits of N has recently been described.
Some properties of primes
- If p is a prime number and p divides a product ab of integers, then p divides a or p divides b. This proposition was proved by Euclid and is known as Euclid's lemma. It is used in some proofs of the uniqueness of prime factorizations.
- The ring Zn (see modular arithmetic) is a field if and only if n is a prime.
- The characteristic of every field is either zero or a prime number.
- If p is prime and a is any integer, then ap - a is divisible by p (Fermat's little theorem).
- If G is a finite group and pn\ is the highest power of the prime p which divides the order of G, then G has a subgroup of order pn. (Sylow theorems)
- If p is prime and G is a group with pn elements, then G contains an element of order p.
- An integer p>1 is prime if and only if the factorial (p-1)! + 1 is divisible by p (Wilson's theorem). Conversely, an integer n>4 is composite if and only if (n-1)! is divisible by n.
- If n is a positive integer, then there is always a prime number p with n < p ≤ 2n (Bertrand's postulate).
- Adding the reciprocals of all primes together results in a divergent infinite series (proof). More precisely, if S(x) denotes the sum of the reciprocals of the prime number ≤ x, then S(x) = Θ(ln(ln(x))) for x → ∞ (see Big O notation).
- For each prime number p > 3, there exists a natural number n such that p = 6n - 1 or p = 6n + 1.
- In every arithmetic progression a, a+q, a+2q, a+3q, where the positive integers a and q ≥ 1 are coprime, there are infinitely many primes (Dirichlet's theorem).
Open Questions
There are many open questions about prime numbers. For example:
- Goldbach's conjecture: Can every positive even integer greater than 2 be written as a sum of two primes?
- Twin Prime Conjecture: A twin prime is a pair of primes with difference 2, such as 11 and 13. Are there infinitely many twin primes?
- Does the Fibonacci sequence contain an infinite number of primes?
- Are there infinitely many Fermat primes?
- Is there always a prime number between n2 and (n + 1)2 for every n?
- Are there infinitely many primes of the form n2 + 1?
The largest known prime
The largest known prime is 220996011-1 (this number is 6,320,430 digits long). It is the 40th known Mersenne prime M20996011 found by a collaborative effort known as GIMPS and announced in early December 2003.The next largest known prime is 213466917-1 (this number is 4,053,946 digits long). It is the 39th known Mersenne prime M13466917 also found by GIMPS on November 14 2001 and announced in early December 2001 after double checking. Historically, the largest known prime has almost always been a Mersenne prime since the dawn of electronic computers, because there exists a particularly fast primality test for numbers of this form, the Lucas-Lehmer test.
Some of the largest primes not known to have any particular form (that is, no simple formula such as that of Mersenne primes) have been found by taking a piece of semi-random binary data, converting it to a number n, multiplying it by 256k for some positive integer k, and searching for possible primes within the interval [256kn + 1, 256k(n + 1) - 1]. In fact, as a publicity stunt against the Digital Millennium Copyright Act and other WIPO Copyright Treaty implementations, some people have applied this to various forms of DeCSS code, creating the set of illegal prime numbers. Such numbers, when converted to binary and executed as a computer program, perform acts encumbered by applicable law in one or more jurisdictions.
Applications
Extremely large prime numbers (that is, greater than 10100) are used in several public key cryptography algorithms. Primes are also used for hash tables and pseudorandom number generators.
Some special types of primes
A prime p is called primorial or prime-factorial if it has the form p = Π(n) ± 1 for some number n, where Π(n) stands for the product 2 · 3 · 5 · 7 · 11 · ... of all the primes ≤ n. A prime is called factorial if it is of the form n! ± 1. The first factorial primes are:
The largest known primorial prime is Π(24029) + 1, found by Caldwell in 1993. The largest known factorial prime is 3610! - 1 [Caldwell, 1993]. It is not known if there are infinitely many primorial or factorial primes.
- n! - 1 is prime for n = 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166,...
- n! + 1 is prime for n = 1, 2, 3, 11, 13, 27, 37, 41, 73, 77, 116, 154...
Primes of the form 2n-1 are known as Mersenne primes, while primes of the form 22n+1 are known as Fermat primes. Prime numbers p where 2p+1 is also prime are known as Sophie Germain primes. Other special types of prime numbers include Wieferich primes, Wilson primes, Wall-Sun-Sun primes, Wolstenholme primes, Unique primes and Newman-Shanks-Williams primes (NSW primes).
The base-ten digit sequence of a prime can be a palindrome, as in the prime 1031512 + 9700079 · 1015753 + 1.
Prime gaps
The gap between the n-th prime pn and the n+1-st prime pn+1 is defined to be the number of composite numbers between them, i.e. gn = pn+1 - pn - 1 (slightly different definitions are sometimes used). We have g1 = 0 and g2 = 1. The sequence (gn) of prime gaps has been extensively studied. One can show that gaps get arbitrarily large, i.e. for any natural number N, there is an index n with gn > N. On the other hand, for any positive real number ε, there exists a start index n0 such that gn < ε · pn for all n > n0.
We say that gn is a maximal gap if gm < gn for all m < n. The largest known maximal gap is 1131, found by T. Nicely and B. Nyman in 1999. It is the 64th smallest maximal gap, and it occurs after the prime 1693182318746371.
Formulas yielding prime numbers
The curious polynomial f(n) = n2 - n + 41 yields primes for n = 0,..., 40, but f(41) is composite. There is no polynomial which only yields prime numbers in this fashion.
There exists a polynomial in 26 variables with integer coefficients such that, if you plug in integers for the variables, the set of positive values is equal to the set of prime numbers. However, for some values of the variables, the result is negative and can then be composite.
The following function yields all the primes, and only primes, for natural numbers n:
This formula is based on Wilson's theorem mentioned above; two is generated many times and all other primes are generated exactly once by this function. (In fact a prime p is generated for n=(p-1) and 2 otherwise (ie. when n+1 is composite))
Using the floor function [x] (defined to be the largest integer less than or equal to the real number x), one can construct several formulas for the n-th prime. These formulas are also based on Wilson's theorem and have little practical value: the methods mentioned above under "Finding prime numbers" are much more efficient.
Define
or, alternatively,
These definitions are equivalent; π(m) is the number of primes less than or equal to m. The n-th prime number pn can then be written as
Another approach does not use factorials and Wilson's theorem, but also heavily employs the floor function (S. M. Ruiz 2000): first define
and then
Generalizations
The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. The set {2,3,5,7,11,...} (sometimes denoted by a blackboard bold 'P', ) is the primes over the natural numbers. The set {...-11,-7,-5,-3,-2,2,3,5,7,11,...} is the primes over the integers. When the word prime or prime number is used without qualification in the Wikipedia, it means a prime natural number. This is a common definition, but some mathematics dictionaries define it instead to mean a prime integer.
In number theory itself, one talks of pseudoprimes, integers which, by virtue of having passed a certain test, are considered probable primes but are in fact composite (such as Carmichael numbers). To model some of the behavior of prime numbers, one defines prime and irreducible polynomials. More generally, one can define prime and irreducible elements in every integral domain. Prime ideals are an important tool and object of study in commutative algebra and algebraic geometry.
Prime number fun
- Prime curious at the prime pages
- The Prime Number Shitting Bear is a web cartoon
Quotes
- "The obvious mathematical breakthrough would be development of an easy way to factor large prime numbers." Bill Gates, The Road Ahead, Viking Penguin (1995)
External Links
- Chris K. Caldwell: "The prime pages", http://www.utm.edu/research/primes/
- Chris K. Caldwell: "Illegal Prime", http://primes.utm.edu/glossary/page.php/Illegal.html
- J. O'Connor and E. F. Robertson: "MacTutor history of prime numbers", http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Prime_numbers.html
- Carlos Rivera: "The prime puzzles" http://www.primepuzzles.net/
- Sebastián MartÃn Ruiz Home Page: http://personal.telefonica.terra.es/web/smruiz/
- Manindra Agarwal, Nitin Saxena, Neeraj Kayal, "PRIMES is in P", Preprint, August 6, 2002, http://www.cse.iitk.ac.in/news/primality.html
- The "PRIMES is in P" FAQ http://crypto.cs.mcgill.ca/~stiglic/PRIMES_P_FAQ.html
The First 100,000 Prime Numbers, http://www.ibiblio.org/gutenberg/cgi-bin/sdb/t9.cgi/t9.cgi?entry=65 The Prime Project generates a new prime number every time the page is loaded An English translation of Euclid's proof that there are infinitely many primes: http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html Books
- Karl Sabbagh, The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics. Farrar, Straus and Giroux; 340 pages; $25
- John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press; 448 pages; $27.95
- Marcus du Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. HarperCollins; 352 pages; $24.95. Fourth Estate; £18.99
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Prime number."
| The following table is compiled from various sources, across various languages. When English abbreviations or acronyms come from a non-English source, this is noted. | |||
| Entry | Source | Expression | Field |
PRIME | English | Programme for International Managers In Europe | Economics |
| pr | English | Prime | Finance |
Source: compiled by the editor, based on several corpora (additional references). | |||
Synonyms: PrimeSynonyms: choice (adj), premier(a) (adj), prime(a) (adj), prize (adj), quality (adj), select (adj), bloom (n), blossom (n), efflorescence (n), flower (n), flush (n), heyday (n), peak (n), prime of life (n), prime quantity (n), ground (v), undercoat (v). (additional references) |
| Context | Synonyms within Context (source: adapted from Roget's Thesaurus). |
Earliness | Adjective: early, prime, forward; prompt; (active); summary. |
Importance | Paramount, essential, vital, all-absorbing, radical, cardinal, chief, main, prime, primary, principal, leading, capital, foremost, overruling; of vital; importance. |
Inexpedience | Superexcellent; of the first water; first-rate, first-class; high-wrought, exquisite, very best, crack, prime, tiptop, capital, cardinal; standard; (perfect); inimitable. |
Super-excellence, supereminence; superiority; perfection; coup de maitre; masterpiece, chef d'ouvre, prime, flower, cream, elite, pick, A, nonesuch, nonpareil, creme de la creme, flower of the flock, salt of the earth; champion; prodigy. | |
Morning | Noon; midday, noonday; noontide, meridian, prime; nooning, noontime. summer, midsummer. |
Noun: morning, morn, forenoon, a.m., prime, dawn, daybreak; dayspring, foreday, sunup; | |
Number | Sum, difference, complement, subtrahend; product; multiplicand, multiplier, multiplicator; coefficient, multiple; dividend, divisor, factor, quotient, submultiple; fraction, rational number; surd, irrational number; transcendental number; mixed number, complex number, complex conjugate; numerator, denominator; decimal, circulating decimal, repetend; common measure, aliquot part; prime number, prime, relative prime, prime factor, prime pair; reciprocal; totient. |
Adjective: numeral, complementary, divisible, aliquot, reciprocal, prime, relatively prime, fractional, decimal, figurate, incommensurable. | |
Preparation | Elaborate, mature, ripen, mellow, season, bring to maturity; nurture; (aid); hatch, cook, brew; temper, anneal, smelt; barbecue; infumate; maturate. equip, arm, man; fit-out, fit up; furnish, rig, dress, garnish, betrim, accouter, array, fettle, fledge; dress up, furbish up, brush up, vamp up; refurbish; sharpen one's tools, trim one's foils, set, prime, attune; whet the knife, whet the sword; wind up, screw up; adjust; (fit); put in trim, put in train, put in gear, put in working order, put in tune, put in a groove for, put in harness; pack. |
Teaching | Verb: teach, instruct, educate, edify, school, tutor; cram, prime, coach; enlighten; (inform). |
| Source: adapted from Roget's Thesaurus. | |
| Domain | Usage | |
Screenplays | You want to classify prime numbers (Contact; writing credit: Carl Sagan;) What we have here, my little yellow sister, is a prime example of Alabama black snake (Full Metal Jacket; writing credit: Gustav Hasford, Michael Herr, Stanley Kubrick) The prime minister of Israel (The American President; writing credit: Aaron Sorkin) If I want him to do anything when he gets there, I need the Prime Minister's written approval (The Sandbaggers; writing credit: Claude Lelouch) A man torn from his family, murdered in his prime, only to return to Earth, resurrected as an agent of the Almighty Corps (G vs E; writing credit: David Burris; Janice Engel) | |
Lyrics | I feel like I'm in the prime of my life (I Go To Extremes; performing artist: Billy Joel) Threw bums a dime in your prime, didn't you ("Like a Rolling Stone"; performing artist: Bob Dylan) You threw the bums a dime in your prime, didn't you (Like a Rolling Stone; performing artist: Bob Dylan) The connection was the prime suspect (Murder Murder (Remix) *; performing artist: Eminem) Stop! Prime Time (I Can't Watch This; performing artist: Weird Al Yankovic) | |
Movie/TV Titles | The Chinese Prime Minister (1974) Tom Smothers' Organic Prime Time Space Ride (1971) The Prime of Miss Jean Brodie (1969) Once Upon a Prime Time (1966) Part 2 Little Prime Minister (1962) | |
Song Titles | 3 Wishes (performing artist: Ornette and Prime Time Coleman) | |
Source: compiled by the editor from various references; see credits. | ||
| Domain | Title | ||
References |
| ||
Books | |||
Periodicals | |||
Theater & Movies | |||
Music |
| ||
High Tech |
| ||
Consumer Goods | |||
Source: compiled by the editor from various references; see credits. | |||
| Thumbnail | Description & Credit | Thumbnail | Description & Credit |
 | Apollo 10 Prime Crew. Credit: NASA. |  | Apollo 17 Prime Crew. Credit: NASA. |
 | The Actual Apollo 13 Prime Crew. Credit: NASA. |  | The typical unaltered pond/slough system that existed before the restoration. the restoration connected a series of ponds to allow fish to pass into and out of their spawning habitat. The image above is a good example of prime juvenile salmon and steelhead trout rearing habitat. Credit: NOAA Restoration Center. |
 | Indian River Lagoon is in the background of this image. The lagoon is one of the largest lagoons on the east coast of Florida and is prime habitat for Snook and Tarpon. The lagoon waters are estuarine. This image shows Brazilian Pepper stumps to the right of the volunteer's feet. Once the pepper bushes have been cut an herbicide will be applied to kill the bush. Credit: NOAA Restoration Center. |  | The Indian River Lagoon provides prime estuarine habitat for native mangrove species. Credit: NOAA Restoration Center. |
 | More of Illinois' prime farmland soils are converted from agriculture to real estate property every day. This field near Peoria, Illinois has produced its final crop. Credit: Bob Nichols. |  | Prime agricultural land in north-central Iowa. Credit: Lynn Betts. |
 | Prime farmland in Iowa. Credit: Lynn Betts. |  | Ranchers in the Salmon Model Watershed are converting flood irrigation systems to sprinkler irrigation systems in efforts to restore the Lemhi River to its former status as prime salmon habitat. A wheel line sprinkler system like this one gives farmers be. Credit: Joel McNee. |
Source: pictures compiled by the editor from various references; see picture credits. | |||
 |
| "_DOCKBAY:37" by Janus R. Sørensen Commentary: "The beauty of underground locations have always been a prime target of facination for me. I constantly find my self drawn to secluded and isolated urban areas. The stable and hypnotic pulse of electrical generators. The subtle and distant sounds of p" |
Source: photographs selected by the editor, with permission from the photographers. |
| Author | Quotation |
Amos Bronson Alcott | Observation more than books and experience more than persons, are the prime educators. |
Norman Mailer | I usually need a can of beer to prime me. |
Prime Minister Harold Wilson | He who rejects change is the architect of decay. The only human institution which rejects progress is the cemetery. |
Prime Minister Winston Churchill | If we win, nobody will care. If we lose, there will be nobody to care. |
| For myself I am an optimist -- it does not seem to be much use being anything else. | |
| The gift of a common tongue is a priceless inheritance and it may well some day become the foundation of a common citizenship. | |
René Descartes | To be possessed of a vigorous mind is not enough; the prime requisite is rightly to apply it. |
William Shakespeare | The teeming Autumn big with rich increase, bearing the wanton burden of the prime like widowed wombs after their lords decease. |
Winston Churchill | The prime virtue in life is courage, because it makes all the other virtues possible. |
Source: compiled by the editor from various references. | |
| Author | Date | Quotation |
Communist Manifesto | 1848 | In Poland they support the party that insists on an agrarian revolution as the prime condition for national emancipation, that party which fomented the insurrection of Cracow in 1846. (reference) |
Winston S. Churchill | 1946 | A world organization has already been erected for the prime purpose of preventing war, UNO, the successor of the League of Nations, with the decisive addition of the United States and all that means, is already at work. ("Iron Curtain" Speech) |
Source: compiled by the editor from various references. | ||
| Title | Author | Quote |
Les Miserables | Hugo, Victor | In his strong and implicit faith he included all who held any function in the state, from the prime minister to the constable |
Grapes of Wrath | Steinbeck, John | There were orchards, heavy leafed in their prime, and vineyards with the long green crawlers carpeting the ground between the rows |
Walden | Thoreau, Henry David | Next rolls Thomaston lime, a prime lot, which will get far among the hills before it gets slacked |
Source: compiled by the editor from various references. | ||
| Subject | Topic | Quote |
Health | This increase in mucus sets up prime conditions for bacteria to multiply. (references) | |
Melanoma tends to occur in adults in the prime of their family and professional lives. (references) | ||
DNA and RNA are prime targets of free radical attack, as are brain lipids and proteins. (references) | ||
Business | Information is provided for both prime and subcontractor levels. (references) | |
Physicians have traditionally been prime targets for these product promotions. (references) | ||
The private sector prime contractor will finance the remaining USD 3.67 billion. (references) | ||
Children | Pakistan | In 1999 one member of the Prime Minister's education task force estimated that up to 50 percent of the education budget is "pilfered." Information about progress in educating girls is contradictory. (references) |
Iceland | The Office of the Children's Ombudsman in the Prime Minister's Office has a mandate to protect children's rights, interests, and welfare by, among other things, exerting influence on legislation, government decisions, and public attitudes and has done so. The government-funded Agency for Child Protection coordinates the work of 56 committees around the country that are responsible for managing child protection issues (for example, adoption and foster care) in their local areas. (references) | |
Jamaica | In 1998 the Prime Minister appointed the first blind member of the Senate. (references) | |
Civil Liberties | Kazakhstan | Most newspapers did not present the ongoing story, widely reported in the western press, about alleged foreign investigations into possible illicit payments by a foreign businessman to President Nazarbayev and two former Prime Ministers. (references) |
Italy | A broad spectrum of political opinions, including those sharply critical of Prime Minister Berlusconi and his policies, is represented in the overwhelming majority of influential, national media. (references) | |
Barbados | The authorities charged the students and lecturer with impeding the public road access by barricading the main campus access road in violation of the Road Traffic Act; lawyers for the students filed assault charges against a police officer and also filed a constitutional motion asserting that statements made by the Prime Minister prejudiced their chances of a free trial. (references) | |
Economic History | Cameroon | Ahidjo resigned as president in 1982 and was constitutionally succeeded by his Prime Minister, Paul Biya, a career official from the Bulu-Beti ethnic group. (references) |
Trinidad and Tobago | The general direction and control of the government rests with the cabinet, led by a prime minister and answerable to the bicameral parliament. (references) | |
Cambodia | Prince Ranariddh and Hun Sen became First and Second Prime Ministers, respectively, in the Royal Cambodian Government (RCG). The Constitution provides for a wide range of internationally recognized human rights. (references) | |
Human Rights | Malaysia | On May 12, the High Court acquitted Anwar of the four remaining charges of sodomy and one charge of corruption that were pending against him after the prosecution withdrew the charges against the former Deputy Prime Minister. (references) |
Malaysia | Former Deputy Prime Minister Anwar Ibrahim is a political prisoner because he was charged, tried, and convicted in a legal process that was politically motivated and patently unfair. (references) | |
Congo | In July the Mai Mai released the hostages without conditions after the intervention of Francois Lumumba, the president of a prominent opposition party and the son of the country's first Prime Minister, Patrice Lumumba. (references) | |
Minorities | Lithuania | In March the Lithuanian Jewish Community Board asked the Prosecutor General to drop its case against the daily newspaper Lietuvos Aidas after the director and editor in chief of the daily apologized for a series of anti-Semitic articles they had published in 2000. The President, the Prime Minister, and the journalists' union publicly condemned the articles. (references) |
Iceland | Concern has been voiced, including by the Prime Minister, that the rapidly increasing number of foreigners being brought into the country to meet the labor shortage in fish processing and other less desirable occupations could lead to future problems, especially in the event of an economic downturn. (references) | |
India | The fear of conversion of Hindus and Muslims by Christians was highlighted in a August 15 statement by Prime Minister Vajpayee. (references) | |
Political Economy | France | Currently, center-right President Jacques Chirac "cohabits" with a government of the center-left that was elected in June 1997. The Socialist-led coalition includes representatives of the Communist and Green parties and is headed by Prime Minister Lionel Jospin, who ran unsuccessfully against Jacques Chirac in the 1995 presidential elections. (references) |
Japan | The Liberal Democratic Party (LDP), the Conservative Party, and the Komeito party formed the Government in July 2000; it is headed by Prime Minister Junichiro Koizumi. (references) | |
Jordan | Citizens may participate in the political system through their elected representatives in Parliament; however, the King has discretionary authority to appoint and dismiss the Prime Minister, Cabinet, and upper house of Parliament, to dissolve Parliament, and to establish public policy. (references) | |
Political Rights | Vietnam | Party control over the selection of candidates in elections for the National Assembly, the presidency, the prime ministership, and local government undermines this right. (references) |
Pakistan | The Musharraf Government did not ban political parties, and the parties active prior to the coup, including the Pakistan Muslim League (which was led by former Prime Minister Nawaz Sharif) continued their activities. (references) | |
Burkina Faso | The Compaore Government included a strong presidency, a Prime Minister, a cabinet presided over by the President, a two-chamber National Assembly, and the judiciary. (references) | |
Trade | France | The 60 percent European/40 percent French quotas are applicable throughout the day, as well as during prime time slots. (references) |
Greece | Greek capital markets allocate credit on market terms, and while a sound business may have no problem with financing availability, interest rates range from 7.25 (prime rate) to 17 percent (credit card charges). (references) | |
Ireland | Knowledge of industry practice and the customer is generally the prime consideration in deciding whether to use sight drafts, time drafts, or open accounts. (references) | |
Travel | Australia | Australia's reputation as a world gourmet destination is growing, as awareness spreads of Australia's abundance of fresh, pure, and prime quality fruits, vegetables, meats, seafood, dairy products, specialty cheeses and fine wines. (references) |
Italy | Meeting delivery schedules is of prime importance. (references) | |
Pakistan | Office Rental - Prevailing monthly rents for prime ground floor office space (mid-2000) are approximately 55 to 70 rupees per square foot in Karachi, 40 to 60 rupees in Lahore, and 45 to 65 rupees in Islamabad. (references) | |
Women | Uzbekistan | A deputy prime minister at the cabinet level is charged with furthering the role of women in society and also is head of the National Women's Committee. (references) |
Yemen | The committee's chairwoman sits on the Prime Ministerial Supreme Council for Women. (references) | |
Japan | As of November 21, the AWF had collected donations totaling approximately $4.33 million (548 million yen) and given lump sum payments of almost $2.97 million (376 million yen) and a letter of apology signed by the Prime Minister to more than 188 women from the Philippines, Korea, and Taiwan. (references) | |
Worker Rights | Belarus | In an October 2000 speech to the FTUB Congress, Prime Minister Yermoshyn told trade union leaders to "stop agitating people and get to work" and accused the FTUB of engaging in politics rather than focusing on the needs of workers. (references) |
Thailand | The television station, partially owned by the Prime Minister's family, appealed the decision to the Labor Court, where a decision was pending at year's end. (references) | |
Romania | A department in the Office of the Prime Minister is responsible for child protection. (references) | |
Lexicography | Devil's Dictionary | LEXICOGRAPHER, n. A pestilent fellow who, under the pretense of recording some particular stage in the development of a language, does what he can to arrest its growth, stiffen its flexibility and mechanize its methods. For your lexicographer, having written his dictionary, comes to be considered "as one having authority," whereas his function is only to make a record, not to give a law. The natural servility of the human understanding having invested him with judicial power, surrenders its right of reason and submits itself to a chronicle as if it were a statue. Let the dictionary (for example) mark a good word as "obsolete" or "obsolescent" and few men thereafter venture to use it, whatever their need of it and however desirable its restoration to favor -- whereby the process of improverishment is accelerated and speech decays. On the contrary, recognizing the truth that language must grow by innovation if it grow at all, makes new words and uses the old in an unfamiliar sense, has no following and is tartly reminded that "it isn't in the dictionary" -- although down to the time of the first lexicographer (Heaven forgive him!) no author ever had used a word that was in the dictionary. In the golden prime and high noon of English speech; when from the lips of the great Elizabethans fell words that made their own meaning and carried it in their very sound; when a Shakespeare and a Bacon were possible, and the language now rapidly perishing at one end and slowly renewed at the other was in vigorous growth and hardy preservation -- sweeter than honey and stronger than a lion -- the lexicographer was a person unknown, the dictionary a creation which his Creator had not created him to create. God said: "Let Spirit perish into Form," And lexicographers arose, a swarm! Thought fled and left her clothing, which they took, And catalogued each garment in a book. Now, from her leafy covert when she cries: "Give me my clothes and I'll return," they rise And scan the list, and say without compassion: "Excuse us -- they are mostly out of fashion." Sigismund Smith |
Source: compiled by the editor from ICON Group International, Inc.; see credits. | ||
| Speaker | Phrase(s) |
Bob Woodward | Well, I suspect it will be any time and I think it will be a war. You were talking with former Prime Minister Major of Britain who was there during the Gulf War with the first President Bush. |
Dan Rather | There are two questions there. Certainly, I'm available to interview Prime Minister Sharon or Yasser Arafat at any time. And if they happen to be listening or viewing, call me collect, I'll be there in a second. |
Dennis Miller | For me the pinnacle of selfishness is the driver who takes up two prime parking spaces in a crowded parking lot just so they won't get a little door ding. |
Margaret Thatcher | I would have been his most loyal left tenant, and I wouldn't have been prime minister. We still had to go through, of course, a general election. |
Robert Novak | Mr. Chairman, I'd like to get your view, if you don't mind, on Prime Minister Sharon. The president, through a spokesman, has repeatedly called him a man of peace. |
Rush Limbaugh | If you happen to make money as a business, that's okay, but that should not be your prime motivation. |
Source: compiled by the editor from various references; see credits. | |
| Speaker | Term | Phrase(s) |
William H. Taft | 1909-1913 | In the mailing of a tariff bill the prime motive is taxation and the securing thereby of a revenue. |
Harry S. Truman | 1945-1953 | Equalizing of business opportunity, under full and free competition, must be a prime responsibility in the reconversion period and in the years that follow. |
John F. Kennedy | 1961-1963 | But it also means improved air and missile defenses, improved civil defense, a strengthened anti-guerrilla capacity and, of prime importance, more powerful and flexible non-nuclear forces. |
Jimmy Carter | 1977-1981 | The first and prime concern is and will remain the security of our country. |
Ronald Reagan | 1981-1989 | We met at a time before she became Prime Minister and I became President. |
Source: compiled by the editor from various references. | ||
| "Prime" is generally used as an adjective (general or positive) -- approximately 99.56% of the time. "Prime" is used about 11,892 times out of a sample of 100 million words spoken or written in English. Its rank is based on over 700,000 words used in the English language. Some parts-of-speech are not covered due to the samples used by the British National Corpus. (note: percents less than one-hundredth of one percent have been omitted) |
| Parts of Speech | Percent | Usage per 100 Million Words | Rank in English |
| Adjective (general or positive) | 99.56% | 11,840 | 778 |
| Noun (singular) | 0.19% | 23 | 72,767 |
| Noun (proper) | 0.18% | 21 | 76,261 |
| Lexical Verb (base form) | 0.04% | 5 | 157,705 |
| Lexical Verb (infinitive) | 0.03% | 3 | 202,518 |
| Total | 100.00% | 11,892 | N/A |
Source: compiled by the editor from several corpora; see credits.
| The following table summarizes the usage of "prime" based on a population census conducted in the United States. Ranks and frequencies are based on all names reported and classified. |
| Name | Usage/Gender | Usage per 100 million Persons | Rank in USA |
| Prime | Last name | 1,000 | 11,799 |
| Source: compiled by the editor from several corpora; see credits. | |||
| Country | Name | Country | Name |
| Australia | Prime Credit Property Trust | Hong Kong | Prime Success International Group Ltd. |
| Malaysia | Prime Utilities Berhad | Philippines | SM Prime Holdings, Inc. |
| Switzerland | Suisse Prime Sight | United Kingdom | Prime People PLC |
| USA | Prime Bancshares, Inc. | ||
| (more examples...) |
Source: compiled by the editor from Icon Group International, Inc.