What is "prime number"

Prime \Prime\, a. [F., fr. L. primus first, a superl. corresponding to the compar. prior former. See Prior, a., Foremost, Former, and cf. Prim, a., Primary, Prince.]

1. First in order of time; original; primeval; primitive; primary. ``Prime forests.''
   --Tennyson.

   She was not the prime cause, but I myself.
   --Milton.

   Note: In this sense the word is nearly superseded by primitive, except in the phrase prime cost.

2. First in rank, degree, dignity, authority, or importance; as, prime minister. ``Prime virtues.''
   --Dryden.

3. First in excellence; of highest quality; as, prime wheat; a prime quality of cloth.

4. Early; blooming; being in the first stage. [Poetic]

   His starry helm, unbuckled, showed him prime In manhood where youth ended.
   --Milton.

5. Lecherous; lustful; lewd. [Obs.]
   --Shak.

6. Marked or distinguished by a mark (') called a prime mark.

7. (Math.)

   1. Divisible by no number except itself or unity; as, 7 is a prime number.

   2. Having no common factor; -- used with to; as, 12 is prime to 25. Prime and ultimate ratio. (Math.). See Ultimate. Prime conductor. (Elec.) See under Conductor. Prime factor (Arith.), a factor which is a prime number. Prime figure (Geom.), a figure which can not be divided into any other figure more simple than itself, as a triangle, a pyramid, etc. Prime meridian (Astron.), the meridian from which longitude is reckoned, as the meridian of Greenwich or Washington. Prime minister, the responsible head of a ministry or executive government; applied particularly to that of England. Prime mover. (Mech.)

      1. A natural agency applied by man to the production of power. Especially: Muscular force; the weight and motion of fluids, as water and air; heat obtained by chemical combination, and applied to produce changes in the volume and pressure of steam, air, or other fluids; and electricity, obtained by chemical action, and applied to produce alternation of magnetic force.

      2. An engine, or machine, the object of which is to receive and modify force and motion as supplied by some natural source, and apply them to drive other machines; as a water wheel, a water-pressure engine, a steam engine, a hot-air engine, etc.

   3. Fig.: The original or the most effective force in any undertaking or work; as, Clarkson was the prime mover in English antislavery agitation.

      Prime number (Arith.), a number which is exactly divisible by no number except itself or unity, as 5, 7, 11.

      Prime vertical (Astron.), the vertical circle which passes through the east and west points of the horizon.

      Prime-vertical dial, a dial in which the shadow is projected on the plane of the prime vertical.

      Prime-vertical transit instrument, a transit instrument the telescope of which revolves in the plane of the prime vertical, -- used for observing the transit of stars over this circle.

n. (context number theory English) Any natural number that is greater than 1 and is divisible only by itself and 1.

n. an integer that has no integral factors but itself and 1

Prime Number published in 1970, is a collection of science fiction stories, written by Harry Harrison.

• "Mute Milton"
• "The Greatest Car in the World"
• "The Final Battle"
• "The Powers of Observation"
• "The Ghoul Squad"
• "Toy Shop"
• "You Men of Violence"
• "The Finest Hunter in the World"
• "Down to Earth"
• "Commando Raid"
• "Not Me, Not Amos Cabot!"
• "The Secret of Stonehenge"
• "Incident in the IND"
• "If"
• "Contact Man"
• "The Pad: a Story of the Day After Tomorrow"
• "A Civil Service Servant"
• "A Criminal Act"
• "Famous First Words"

A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example, 5 is prime because 1 and 5 are its only positive integer factors, whereas 6 is composite because it has the divisors 2 and 3 in addition to 1 and 6. The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 can be expressed as a product of primes that is unique up to ordering. The uniqueness in this theorem requires excluding 1 as a prime because one can include arbitrarily many instances of 1 in any factorization, e.g., 3, 1 · 3, 1 · 1 · 3, etc. are all valid factorizations of 3.

The property of being prime (or not) is called primality. A simple but slow method of verifying the primality of a given number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and $\sqrt{n}$. Algorithms much more efficient than trial division have been devised to test the primality of large numbers. These include the Miller–Rabin primality test, which is fast but has a small probability of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as Mersenne numbers. , the largest known prime number has 22,338,618 decimal digits.

There are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no known simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, the statistical behaviour of primes in the large, can be modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability that a given, randomly chosen number is prime is inversely proportional to its number of digits, or to the logarithm of n.

Many questions regarding prime numbers remain open, such as Goldbach's conjecture (that every even integer greater than 2 can be expressed as the sum of two primes), and the twin prime conjecture (that there are infinitely many pairs of primes whose difference is 2). Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which makes use of properties such as the difficulty of factoring large numbers into their prime factors. Prime numbers give rise to various generalizations in other mathematical domains, mainly algebra, such as prime elements and prime ideals.