Prime number

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 . 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. As of January 2016, the largest known prime number has 22,338,618 decimal digits.

Thirteen (Transformers)

The Thirteen Original Transformers (sometimes referred to as the Thirteen Primes, the Dynasty of Primes or simply the Thirteen) are thirteen fictional characters in the Transformers toy and entertainment franchise created by Hasbro. They are usually portrayed as the first thirteen living beings created by Primus, the creator of the Transformers and their home planet of Cybertron. In the live-action Transformers film trilogy, their back-story is slightly modified, with there only being seven members who were created by the Allspark. The story concept of the Thirteen, as well as three of their members (Prima, Liege Maximo and The Fallen), were created by comic book writer Simon Furman.

Concept history

It was originally revealed in the Botcon story "Covenant" that before going on to create the Transformer race, Primus experimented with a group of twelve robots known as the Covenant of Primus. Based on the zodiac, these twelve robots were: Leonicus (leader), Aries, Piscor, Libras, Scorpius, Aquator, Taurotron, Cancix, Geminator, Sagittarii, Virgol and Capricurn . They existed in secret, training for the Omega Point. The Omega Point was a potential event that Unicron would attack at a point that there was no Prime. After the Omega Point, time was reverted and the instance never happened.

Primality test

A primality test is an algorithm for determining whether an input number is prime. Amongst other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy (its running time is polynomial in the size of the input). Some primality tests prove that a number is prime, while others like Miller–Rabin prove that a number is composite. Therefore, the latter might be called compositeness tests instead of primality tests.

Simple methods

The simplest primality test is trial division: Given an input number n, check whether any prime integer m from 2 to √n evenly divides n (the division leaves no remainder). If n is divisible by any m then n is composite, otherwise it is prime.

For example, we can do a trial division to test the primality of 100. Let's look at all the divisors of 100:

Doctor

Doctor or The Doctor may refer to:

Personal titles

Fictional characters

Film and television

Doctors (TV series)

Doctors is a British medical soap opera which first broadcast in the United Kingdom on BBC One on 26 March 2000. Set in the fictional Midlands town of Letherbridge, defined as being in the city of Birmingham, the soap follows the staff of a doctor's surgery, and their families.

History

Doctors is produced by BBC Birmingham and is screened on BBC One, with the first episode broadcast on 26 March 2000. It was created by Chris Murray, with Mal Young developing it and Carson Black the original producer. The show has been shown at lunchtime since its inception, originally at 12.30pm as a lead-in to the BBC's One O'Clock News. After it was temporarily moved to allow for extended news coverage of the 11 September 2001 attacks, its regular slot changed to 2:10 pm, following directly after Neighbours, after ratings rose to a 25% audience share. When the BBC lost Neighbours to Channel 5 in January 2008, it moved into the Australian soap's old slot of 1:45pm. For a brief trial period in Summer 2000, selected episodes from the first series were shown on Fridays at 7 pm and from 16 February 2009, the show began transmitting in high definition on BBC HD at 4:00pm the same day.

Doctors (series 9)

Series 9 of the long-running medical soap opera Doctors commenced on 23 April 2007. This series consisted of 212 episodes and concluded on 20 March 2008.

Cast

Main characters

Recurring characters

Crew

Episodes

References