Euclid's original proof adds a third: the two lengths are not prime to one another.

Although not officially announced, the original backup crew consisted of Fred Haise ( CDR ), William R. Pogue ( CMP ) and Gerald Carr ( LMP ) who were targeted for the prime crew assignment on Apollo 19.

The simplest, and original, implementation of the protocol uses the multiplicative group of integers modulo p, where p is prime and g is primitive root mod p. Here is an example of the protocol, with non-secret values in < span style =" color: blue "> blue </ span >, and secret values in < span style =" color: red "> boldface red </ span >:

In number theory, integer factorization or prime factorization is the decomposition of a composite number into smaller non-trivial divisors, which when multiplied together equal the original integer.

By the 1920s however the Senate had lost much of its original influence, and hence no sitting senator had been known to have serious aspirations of becoming prime minister whilst remaining in the Senate.

The original remuneration for prime minister and other ministers were specified in the Part B of the second schedule of the constitution of India, which was later removed by an amendment.

Referring to the book's original 1987 publication in a Rolling Stone review pegged to the book's republication in 2002, the critic Jon Caramanica wrote, " It might well be the most comprehensive biography ever written about a pop act while it was still in its prime.

The original prime contractor and developer of the missile was Lockheed Martin Space Systems.

Fox's original reason for the reduced number of prime time hours was to avoid fulfilling the FCC's requirements at the time to be considered a network, and to be free of resulting regulations, though FCC rules have been relaxed since then.

" In it he proves in an original manner the theorem of Fermat ---" That every prime number of the form ( being an integer ) is the sum of two square numbers.

Encore presentations of Disney Channel Original Movies however, still sometimes air on the channel in prime time on the Saturday and Sunday after its original Friday night debut ( Camp Rock, Dadnapped, Starstruck, and Harriet the Spy: Blog Wars are the only exceptions to this rule ).

The show's popularity allowed for two spin-off series: Saved by the Bell: The College Years ( 1993 – 1994 ), a prime time series which followed several of the original characters ' college experiences, and Saved by the Bell: The New Class ( 1993 – 2000 ), a Saturday-morning series which followed a new group of students at Bayside High School.

The residential area is split into two sections known as the village and the town, with the village, near the common, centred around the High street being part of the original medieval village, and now a prime residential area of London commanding high prices, and the " town " being part of the modern development, centred around The Broadway, since the building of the railway station in 1838.

The town quickly spread out from the original Hickory Valley location, its population growth fueled successively by the town becoming the county seat in 1821, the establishment of a railroad line connecting the city with Columbia and Charleston in the 1880s, the establishment of an airfield in the 1930s and more recently the establishment of Interstate 95 in the 1960s, making the town a prime overnight stop on the road to Florida or New York.

Like the original Riemann hypothesis, it has far reaching consequences about the distribution of prime numbers.

In modern editions of Western classical works, editors often seek to eliminate the potential for different interpretations of ornamental symbology, of which grace notes are a prime example, by converting a composer's original ornamental notation into literal notation, the interpretation of which is far less subject to variation.

* September 13 – U. S. network prime time programming shrinks as the original Prime Time Access Rule takes effect.

All these conjectures can be formulated for more general fields in place of Q: algebraic number fields ( the original and most important case ), local fields, and function fields ( finite extensions of F < sub > p </ sub >( t ) where p is a prime and F < sub > p </ sub >( t ) is the field of rational functions over the finite field with p elements ).

In 2008 Bochco argued that the new home for quality prime time drama is cable, where " the atmosphere is far friendlier and the creative environment more conducive to doing original work ", and that " most of what's passing for primetime drama these days isn't very good.

The original prime crew of cosmonauts for Voskhod 1, composed of Boris Volynov, Georgi Katys, and Boris Yegorov, was rejected just three days before the scheduled launch date for the space capsule, reportedly due to the State Commission's discovery that Volynov's mother was Jewish.

The original prime crew for Soyuz 11 consisted of Alexei Leonov, Valeri Kubasov and Pyotr Kolodin.

According to the fundamental theorem of arithmetic, any number obtained in this way can be uniquely factored into prime factors, so it is possible to recover the original sequence from its Gödel number ( for any given number n of symbols to be encoded ).

The original prime crew for Mercury Atlas-7 was to have been Deke Slayton, with Schirra as his back-up.

Another way, perhaps more similar to the original, to interpret the modern definition is to realize that the elements of A can actually be thought of as functions on the prime ideals of A ; namely, as functions on Spec A.

The Sazerac is a local New Orleans variation of an old-fashioned cognac or whiskey cocktail, named for the Sazerac de Forge et Fils brand of cognac that was its original prime ingredient.

Though they used the original Japanese show as the medium for the dub, some of the voices seem to sound like the U. S. version ( e. g., Taichi having an adolescent's voice instead of a kid's ) or completely original to the dubbing crew ( e. g., Gabumon's deep, grumbly voice ).

* Supplements: Audio commentary by producer Paul M. Heller and screenwriter Michael Allin, Isolated music score, an all-new introduction and interview with Linda Lee Cadwell, " Location: Hong Kong with Enter the Dragon " original 1973 documentary, " Backyard Workout with Bruce ", " Bruce Lee In His Own Words ", theatrical trailers, TV spots, cast and crew biographies, " Significance of Belts in Martial Arts " notes, " Heir to the Throne " – Jackie Chan notes, retrospective of Hong Kong martial arts films notes and stills, behind-the-scenes notes, reel recommendations – 16 movies

* Supplements: Audio commentary by producer Paul M. Heller and screenwriter Michael Allin, isolated music score, an all-new introduction and interview with Linda Lee Cadwell, " Location: Hong Kong with Enter the Dragon " original 1973 documentary, " Backyard Workout with Bruce ", " Bruce Lee: In His Own Words ", theatrical trailers, TV spots, cast and crew biographies, 10 exclusive Enter the Dragon postcards, 8 reproductions of original lobby cards, reproduction of the original press brochure

Details of Magellan's visit, and the first known Western documentation of Guam and the Chamorro, come from the journal of Antonio Pigafetta, one of Magellan's 18 original crew members to survive Ferdinand Magellan's ill-fated circumnavigation of the globe.

The original colonists signed the " Covenant of Planetfall ", agreeing that this outcome was just recompense for the labors of the crew during the voyage ; that they signed at gunpoint as they were awakened from hibernation is kept secret from later generations, and also that those who refused, died.

The Mars Society and Stanford studies retain the original 2-vehicle mission profile of Mars Direct, but increase the crew size to six.

Yoshiyuki Tomino's original plot for the anime was considerably much more grim, with Amuro dying halfway through the series, and the crew of the White Base having to ally with Char ( who is given a red Gundam ), but finally having to battle him after he takes control of the Principality of Zeon.

* In Die Erde is nah: die Marsexpedition ( 1970 ), by Ludek Pesek, published in English as The Earth is Near ( 1973 ), a crew of twenty men undertake the first manned mission to Mars, and after landing in the wrong area due to computer malfunction and / or human error, have to cope with vicious sand storms, and a long trek undertaken in vehicles to their original intended area of exploration.

The success in syndication of the original live action series and fan pressure for a Star Trek revival led to The Animated Series from 1973 – 1974, as the source of new adventures of the Enterprise crew, the next being the 1979 live-action feature film Star Trek: The Motion Picture.

* Star Trek: Vulcan's Soul is a trilogy written by Sherman and Shwartz between 2004 and 2007 ; set in the aftermath of the Dominion War, it focuses on members of Kirk's original crew becoming involved in a war between the Romulans and a fellow Vulcan off-shoot, the Watraii.

This ship first appears at the conclusion of Star Trek IV: The Voyage Home ( 1986 ) and is the main setting in the following Star Trek movies which use the original crew.

The season ended in May 1998 with " The End ", the final episode shot in Vancouver and the final episode with the involvement of many of the original crew members who had worked on the show for its previous five years, including director and producer R. W.

With the move to Los Angeles, California in season six, many changes behind the scenes occurred, as much of the original X-Files crew was gone.

Many of the descendants of the original crew and colonists are unaware, however, that they are aboard a ship.

Another theory, advanced by the original crash investigator in 2005, hypothesizes that a cabin air vent was accidentally left open by the crew or the previous pilot, leading to oxygen deprivation and leaving the crew incapable of controlling the aircraft.

Moe quickly signed movie and burlesque comic Joe DeRita for the " third Stooge " role ; DeRita adopted first a crew cut and then a completely shaven hairstyle and became " Curly Joe " because of his resemblance to the original Curly Howard ( also to make it easier to distinguish him from Joe Besser, the earlier Stooge called Joe ). The Three Stooges with Curly Joe DeRita filling the role of the third stooge.

Polenin, the original captain, and the crew have served together for some time but Vostrikov's appointment is alleged to have been aided by his wife's political connections ( her uncle is a member of the Politburo ).

The producers made some efforts to work with the original crew of K-19, who took exception to the first version of the script available to them.

A member of the original crew is killed upon landing on a primitive planet, Andrecia, when she is shot at.