In number theory, a Carmichael number is a composite number n which satisfies the modular arithmetic congruence relation: b n − 1 ≡ 1 for all integers b which are prime to n. They are named for Robert Carmichael; the Carmichael numbers are the subset K1 of the Knödel numbers. Equivalently, a Carmichael number is a composite number n. Fermat's little theorem states that if p is a prime number for any integer b, the number bp − b is an integer multiple of p. Carmichael numbers are composite numbers which have this property. Carmichael numbers are called Fermat pseudoprimes or absolute Fermat pseudoprimes. A Carmichael number will pass a Fermat primality test to every base b prime to the number though it is not prime; this makes tests based on Fermat's Little Theorem less effective than strong probable prime tests such as the Baillie–PSW primality test and the Miller–Rabin primality test. However, no Carmichael number is either an Euler–Jacobi pseudoprime or a strong pseudoprime to every base prime to it so, in theory, either an Euler or a strong probable prime test could prove that a Carmichael number is, in fact, composite.

Arnault gives a 397-digit Carmichael number N, a strong pseudoprime to all prime bases less than 307: N = p ⋅ ⋅ where p = 29674495668685510550154174642905332730771991799853043350995075531276838753171770199594238596428121188033664754218345562493168782883is a 131-digit prime. P is the smallest prime factor of N, so this Carmichael number is a pseudoprime to all bases less than p; as numbers become larger, Carmichael numbers become rare. For example, there are 20,138,200 Carmichael numbers between 1 and 1021. An alternative and equivalent definition of Carmichael numbers is given by Korselt's criterion. Theorem: A positive composite integer n is a Carmichael number if and only if n is square-free, for all prime divisors p of n, it is true that p − 1 ∣ n − 1, it follows from this theorem that all Carmichael numbers are odd, since any composite number, square-free will have at least one odd prime factor, thus p − 1 ∣ n − 1 results in an dividing an odd, a contradiction. From the criterion it follows that Carmichael numbers are cyclic.

Additionally, it follows that there are no Carmichael numbers with two prime divisors. Korselt was the first who observed the basic properties of Carmichael numbers, but he did not give any examples. In 1910, Carmichael found the first and smallest such number, 561, which explains the name "Carmichael number"; that 561 is. Indeed, 561 = 3 ⋅ 11 ⋅ 17 is square-free and 2 ∣ 560, 10 ∣ 560 and 16 ∣ 560; the next six Carmichael numbers are: 1105 = 5 ⋅ 13 ⋅ 17 1729 = 7 ⋅ 13 ⋅ 19 2465 = 5 ⋅ 17 ⋅ 29 2821 = 7 ⋅ 13 ⋅ 31 6601

KFIR is a News/Talk radio station with new studios in Albany, OR 917 SE 19th Ave Albany, OR 97322 in the U. S. state of Oregon. KFIR serves Albany, Corvallis and Salem, all in the Willamette Valley. Radio Fiesta Network, LLC, owns the news and talk station.720 AM is a United States clear-channel frequency on which there are nine radio stations in the United States, four in Mexico and none in Canada. KOTZ in Kotzebue, Alaska and WGN in Chicago, Illinois are the dominant Class A stations. Query the FCC's AM station database for KFIR Radio-Locator Information on KFIR Query Nielsen Audio's AM station database for KFIR

The Choice Music Prize, known for sponsorship reasons as the RTÉ Choice Music Prize is an annual music prize awarded to the best album from a band or solo musician who resides in the Republic of Ireland and/or Northern Ireland. Since it first began in 2005, the main awards ceremony had been broadcast live on the Irish independent and national radio station, Today FM, every March with the exception of the 2014 ceremony which took place on February 27, 2014 and it is held in Vicar Street, Dublin with the exception of the 2012 ceremony, held in the Olympia Theatre. After being broadcast on Today FM for nearly eleven years, in November 2016, it was announced that the Choice Music Prize would broadcast on RTÉ 2FM starting in 2017. Previous presenters of the main awards ceremony have been Michelle Doherty and Today FM radio presenters Alison Curtis and Paul McLoone; the program has been sponsored since 2011 by Meteor and it was broadcast live on Today FM since it first began in 2005, airing as a four-part special, beginning at 7pm and concluding at 11pm.

The Choice Music Prize was established by journalist Jim Carroll and manager Dave Reid in 2005. According to organisers, the aim of the Choice Music Prize, is to get more airplay for Irish acts both domestically and overseas. "It was a strange thing" remembers Cormac Brady of Super Extra Bonus Party. "It's not something we expected to happen to us. Winning awards wasn't what we got into music for, but it brought us a hell of a lot more recognition overnight and opened a lot of doors". Julie Feeney concurs. "It was the biggest achievement of my life" she says. "It meant a phenomenal amount to me. It was an enormous validation" as an alternative to the industry-dominated Meteor Music Awards; the Choice Music Prize is modelled after the Mercury Prize, awarded each year to the best album from the United Kingdom or Ireland. It is a music award voted for by a panel of twelve judges based on artistic merit, regardless of genre, sales, or record label; the price includes a €10,000 cheque jointly funded by the Irish Music Rights Organisation and the Irish Recorded Music Association.

There is no sponsorship. Considered by co-founder Carroll as a "titchy little maverick event", the Choice Music Prize has gained a reputation for producing "unpredictable" winners. Winners thus far consist of four bands and one solo male performer. Julie Feeney won the inaugural prize for Irish Album of the Year 2005, she was followed by The Divine Comedy, Super Extra Bonus Party and Jape, winners of Irish Album of the Year 2006, 2007 and 2008 respectively. Adrian Crowley won Irish Album of the Year 2009, while Two Door Cinema Club won Irish Album of the Year 2010; the ceremony to announce the winner takes place at Vicar Street, Dublin in February or March each year. Presented by Michelle Doherty and Rigsy, by Alison Curtis. Today FM radio presenter Paul McLoone presented the awards since 2012, having made his debut presiding over the 2011 Choice Music Prize ceremony and was the current host or M. C.- master of ceremonies for the prize giving ceremony with the show being broadcast live since its inception on the national and independent radio station Today FM as part of a four-hour special, airing between 7pm to 10pm.

In November 2016, the Choice Music Prize announced that they had partnered with RTÉ and as such, the live ceremony will be broadcast on RTÉ 2FM from 2017 onwards. The nominated acts are invited to perform in front of a live audience at the ceremony. However, some nominated acts, such as The Chalets, Fionn Regan, Snow Patrol, Lisa Hannigan, Bell X1 and Laura Izibor, have not performed in the past due to other commitments. David Holmes and The Script did not perform when nominated, though Holmes and Danny O'Donoghue attended the ceremony; the judging panel, composed of various members of Irish media as such as music and broadcasting are all locked in an enclosed room during the performances on the night to debate over which act ought to win with Irish journalist Tony Clayton Lea who works for the Irish Times the Chairman of the Judging Panel with Clayton Lea tasked with helping the panel come to a decision where they all select the one musician or band who will be announced as the winner of the Prize.

A secret ballot was used to decide the winner of Irish Album of Ritual. Past winners and nominees have credited the Choice Music Prize with boosting their careers. Julie Feeney described winning Irish Album of the Year 2005 as "probably the biggest achievement of my life", adding "it meant a phenomenal amount to me, it was an enormous validation". Duke Special, nominated for the first two awards, said his nomination for the inaugural award had helped raise his profile in the Irish media. Cormac Brady stated Super Extra Bonus Party's Irish Album of the Year 2007 win "brought us a hell of a lot more recognition overnight and opened a lot of doors". Nominees have trebled sales after the award has been announced. Steve Jordan was influenced by the Choice Music Prize. Culture Ireland invited figures from the international music industry to the event that decided the Irish Album of the Year 2010. From its inception, the awards show was broadcast live on Today FM and a live streamed on entertainment.ie.

In 2017, the awards show moved to a live broadcast on RTÉ 2FM, highlights will continue to be shown on RTÉ2. The event was aired live each year on Today FM in a special awards ceremony that takes place at Vicar Street, presented by Paul McLoone and a pre-recorded show is aired on RTÉ2 television since 2015 on hosted by Bláthnaid Treacy; the event used to be streamed live on enterta

Lithuania returned to the Eurovision Song Contest in 1999, having withdrawn after their debut entry in 1994. Lithuania's second Eurovision entry came from Aistė, who performed "Strazdas" at the 1999 Contest in Jerusalem, Israel. For Lithuania's return to Eurovision after five years, the Lithuanian broadcaster Lietuvos radijas ir televizija held a national final to select the second Lithuanian entry; the contest was held on 27 December 1998, but was not broadcast on television until 31 December, was recorded in the capital Vilnius, hosted by Vilija Grigonytė and Vytautas Kernagis. Both televoting and jury voting was used during the show, however only the votes of the jury decided the winner; the televote results showed that RebelHeart had won with 40%, with Aistė Smilgevičiūtė coming second with 33%. The winner was decided with only the top 4 songs being known; the winner of the jury vote was Aistė with "Strazdas", performed in Samogitian, a dialect of Lithuanian. Aistė opened the 1999 contest, performing first in a field of 23, preceding Belgium in the running order.

"Strazdas" received 13 points: 1 point from Malta, two points each from Estonia and Croatia, three points from Israel, 5 points from Cyprus, placing 20th in a field of 23. As such Lithuania was relegated from the 2000 Contest, due to low average scores over the past five years. Eurovision Song Contest 1999 Lithuania in the Eurovision Song Contest Lithuanian National Final 1999

A Great Way To Care is a Hong Kong medical crime drama television series produced by Marco Law and TVB. It stars Alex Fong as Dr. Ko Lap-yan, a senior psychiatrist who has a specialty in diagnosing mentally ill criminals. Produced in 2008, the first series premiered on Malaysia's AOD on 8 June 2009. Scheduled for a summer 2009 release in Hong Kong, the broadcast of the series was put on hold after the success of Lee Tim-sing's period serial, Rosy Business. Lee's sister production, Sweetness in the Salt, was aired in place of A Great Way to Care. With a total of 20 episodes, the series premiered on Jade in Hong Kong on 10 January 2011. A sequel series of 25 episodes was renewed in early 2012; the sequel will premiere on TVB's Jade and HD Jade channels on 18 March 2013. Dr. Ko Lap-yan heads a team of senior psychiatrists and residents at the Yan Wo Hospital in Hong Kong. Most episodes revolve around the team diagnosing patients. With a specialty at diagnosing criminals, Ko has close ties with the Serious Crime Unit of the Hong Kong Police Force, a unit headed by Senior Inspector Chung Kwon-ban.

The second series sees Ko's resignation from the Yan Wo Hospital. Season 2 Nominated: Best Drama Nominated: Best Supporting Actor Nominated: Best Supporting Actress Malaysia - 8TV from II onwards. TVB.com A Great Way to Care - Official Website

This is a list of villages in Massachusetts, arranged alphabetically. In Massachusetts, villages do not have any official legal status; the terms "community", "district", "neighborhood", "section" are used to describe these non-municipal entities, which vary in size and relative geographic isolation. Boundaries are sometimes ambiguous, though the United States Census uses the term census-designated place when it assigns boundaries to these entities for the purpose of tabulating community demographics. Many villages have neighborhood associations; the following names are currently or were used to describe places in Massachusetts: List of municipalities in Massachusetts Category:Census-designated places in Massachusetts Turners Falls Montague Franklin Archaic Community, Neighborhood Section and Village, Names in Massachusetts, Secretary of the Commonwealth of Massachusetts