Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states: Every integer greater than 2 can be expressed as the sum of two primes; the conjecture has been shown to hold for all integers less than 4 × 1018, but remains unproven despite considerable effort. A Goldbach number is a positive integer that can be expressed as the sum of two odd primes. Since 4 is the only number greater than 2 that requires the prime 2 in order to be written as the sum of two primes, another form of the statement of Goldbach's conjecture is that all integers greater than 4 are Goldbach numbers; the expression of a given number as a sum of two primes is called a Goldbach partition of that number. The following are examples of Goldbach partitions for some numbers: 6 = 3 + 3 8 = 3 + 5 10 = 3 + 7 = 5 + 5 12 = 7 + 5... 100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53... The number of ways in which 2n can be written as the sum of two primes is: 0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 4, 4, 2, 3, 4, 3, 4, 5, 4, 3, 5, 3, 4, 6, 3, 5, 6, 2, 5, 6, 5, 5, 7, 4, 5, 8, 5, 4, 9, 4, 5, 7, 3, 6, 8, 5, 6, 8, 6, 7, 10, 6, 6, 12, 4, 5, 10, 3....

On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler, in which he proposed the following conjecture: Every integer that can be written as the sum of two primes, can be written as the sum of as many primes as one wishes, until all terms are units. He proposed a second conjecture in the margin of his letter: Every integer greater than 2 can be written as the sum of three primes, he considered 1 to be a prime number, a convention subsequently abandoned. The two conjectures are now known to be equivalent, but this did not seem to be an issue at the time. A modern version of Goldbach's marginal conjecture is: Every integer greater than 5 can be written as the sum of three primes. Euler replied in a letter dated 30 June 1742 and reminded Goldbach of an earlier conversation they had, in which Goldbach remarked his original conjecture followed from the following statement Every integer greater than 2 can be written as the sum of two primes,which is, thus a conjecture of Goldbach.

In the letter dated 30 June 1742, Euler stated: "Dass … ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses theorema, ungeachtet ich dasselbe nicht demonstriren kann." Goldbach's third version is the form in which the conjecture is expressed today. It is known as the "strong", "even", or "binary" Goldbach conjecture, to distinguish it from a weaker conjecture, known today variously as the Goldbach's weak conjecture, the "odd" Goldbach conjecture, or the "ternary" Goldbach conjecture; this weak conjecture asserts that all odd numbers greater than 7 are the sum of three odd primes and appears to have been proved in 2013. The weak conjecture is a corollary of the strong conjecture: if n – 3 is a sum of two primes n is a sum of three primes; the converse implication and the strong Goldbach conjecture remain unproven. For small values of n, the strong Goldbach conjecture can be verified directly. For instance, Nils Pipping in 1938 laboriously verified the conjecture up to n ≤ 105.

With the advent of computers, many more values of n have been checked. One record from this search is that 3325581707333960528 is the smallest number that has no Goldbach partition with a prime below 9781. Statistical considerations that focus on the probabilistic distribution of prime numbers present informal evidence in favour of the conjecture for sufficiently large integers: the greater the integer, the more ways there are available for that number to be represented as the sum of two or three other numbers, the more "likely" it becomes that at least one of these representations consists of primes. A crude version of the heuristic probabilistic argument is as follows; the prime number theorem asserts that an integer m selected at random has a 1 / ln ⁡ m chance of being prime. Thus if n is a large integer and m is a number between 3 and n/2 one might expect the probability of m and n − m being prime to be 1 /. If one pursues this heuristic, one might expect the total number of ways to write a large integer n as the sum of two odd primes to be ∑ m = 3 n / 2 1 ln ⁡ m 1 ln ⁡ ≈ n 2 2.

Since this quantity goes to infinity as n increases, we expect that every large integer has not just one repre

Some Voices is a 2000 British drama film directed by Simon Cellan Jones and adapted for the screen by Joe Penhall, from his own stage play. It is the first feature film by Cellan Jones, a renowned TV director respected for his work on the BAFTA-winning Our Friends in the North; the film was entirely shot on location in Shepherd's Bush, West London, where Cellan Jones lives. It has a running time of 101 minutes; the film's central character, has schizophrenia. The story begins with Ray's discharge from psychiatric hospital. Ray's devoted brother Pete picks him up and drives Ray to his new abode, the spare room in Pete's West London flat. Pete is a chef, he now has to find the time to take care of Ray and monitor the medication that controls the voices in his head. Ray is an out-going young man, he soon falls for a Glaswegian girl in the midst of breaking up with her abusive boyfriend. Laura becomes attracted to his childlike sense of fun. Around this time, Pete becomes involved in a relationship with Mandy.

As Ray's relationship blossoms, he begins to resent taking his pills, preferring to trust in the soothing properties of love. Over time, this decision has disastrous effects on all three relationships: the relationship between the brothers and Laura, Pete and Mandy. Ray may cause disruption and distress to those close to him but, only a fraction of the distress his condition causes him. In the end, it is the relationship between the brothers, central to the film. Pete is long-suffering but, despite all his frustration and resentment, his loving commitment keeps his brother from serious harm. Daniel Craig – Ray Kelly Macdonald – Laura David Morrissey – Pete Julie Graham – Mandy Peter McDonald – Dave Nicholas Palliser – Friend Edward Tudor-Pole - Lighter seller "Speed of the Sound of Loneliness" – Alabama 3 "Rake It In" – Imogen Heap "This Is the Tempo" - Grand Theft Auto "54-46 Was My Number" – Toots and the Maytals "Goodbye Girl" – Squeeze "Il ragazzo della Via Gluck"" - Françoise Hardy Some Voices premiered in Directors' Fortnight at Cannes.

It was nominated for the Golden Hitchcock Award at the annual Dinard Festival of British Cinema in France. The first-time director was nominated for the Best Newcomer at the British Academy Awards. Critics were divided about Some Voices. Channel 4 called it "one of the best British films of 2000", with the director's vision of west London's "tower blocks, dual-carriageways and crowded streets" mirroring the central character's "gradual disintegration". Mark Wyman of Film Review recommended Some Voices as a film "definitely worth seeing" which "showcases some terrific British talent". Conversely, Empire called the film, "claustrophobic and cornered" and claimed that it "probably brought the house down on stage, but on film, it's static". Total Film had doubts. "Perhaps it's the quirky, jerky This Life camerawork or the dim, grainy film stock, but Some Voices never reaches out and grabs the audience, remaining a watch rather than an experience". However, the director did "draw intelligent, effective turns from his cast – Daniel Craig and David Morrissey are excellent, while Kelly Macdonald delivers the kind of sweetly sexy performance she's trade-marking".

Special mention was made of one particular "brilliant effect". As Ray, Daniel Craig's central character, "stops taking his tablets, the strange staticky images start to dominate his vision, the odd sounds begin to blot out reality and the gulf between the world he experiences and the one everyone else lives in widens disastrously. It's a clever and mammothly effective technique, communicating not just the strangeness of what's happening to Ray, but the sheer terror of it". Time Out, on the other hand, described this "brilliant effect" as "over-egging it somewhat"; the "whirling camera effects and freaky sound mix overstates the point that our man is not well". The Time Out reviewer is complimentary about the acting. "Penhall's adaptation of his play remains an actors' showcase. Morrissey skillfully registers abiding filial love tested by simmering exasperation. Peter Bradshaw in The Guardian and Peter Byrne in the Student BMJ are much more unequivocal in their praise. Bradshaw calls the film "a serious and compassionate movie which demands to be seen, not least for its outstanding performances from an excellent cast".

Byrne says that Some Voices "works as a film, a technically accomplished one at that". Both reviewers commend the film's avoidance of cliché. Bradshaw stresses that "nothing could be more tiresome and dishonest than shop worn RD Laing-style clichés about schizophrenia being a heightened visionary state which the western world crushes under the jackboot of its dull rationalist enlightenment; such a proposition would not correspond to the actual experience of schizophrenia sufferers and their carers. He welcomes the fact that "the schizophrenic is not demonised as a potential criminal or as a care-in-the-community basket-case", that "Ray's essential humanity is transcribed with sympathy and warmth, so is the patience and perse

Marriage promotion is a neoliberal policy aiming to produce "strong families" for the purposes of social security. The George W. Bush Administration had focused on government marriage promotion as the solution to the high poverty rates experienced by single-parent families, diverting to marriage promotion tens of millions of dollars appropriated by Congress for other purposes, lobbying Congress to establish a federal marriage promotion program; these programs seek to get unmarried parents to deter separation or divorce. One of the earliest known marriage promotion laws, the Lex Papia Poppaea, imposed penalties on those who refused to get married before a certain age. Provisions against adultery were made in this law. Caelibes could not take a legacy. If he did not comply with the law, the gift became caducum. Marriage promotion includes laws, budget allocations, administrative regulations, think-tank recommendations, operating programs that work in the favor of married people while disfavoring unmarried people.

Heterosexual couples are told to enter and stay in government-certified marriages in order to be economically and responsible citizens. This concept less coincides with the concept of a covenant marriage. Same-sex marriage and cohabitation are ignored by all marriage promotion programs due to moral reasons; this promotion has its roots in the roots in the 1996 Welfare Reform Act. This legislation permitted American states to deny assistance to qualified applicants - resulting in the abrogation of some applicants' constitutional rights. Childbirth with marriage is supported along with the marriage promotion as two people can raise a baby better than an unwed mother or father. Marriage was promoted in the 1990s. Rising divorce rates in the 1980s and 1990s in addition to plummeting marriage rates, allowed then-current U. S. President George W. Bush to pass a nationwide marriage promotion law in the 2000s. A major impetus behind marriage promotion is increasing father involvement. Low-income fathers are forced to take more responsibility for childrearing and their relationships with female partners.

From a starting point of underfunded schools and family chaos, they do poorly in school and drop out. Fathers are urged to get married to the women that they impregnate so that they can establish traditional families, according to the Alliance for Marriage. Marriage promotion may lead to discrimination against single-parent families that increases their poverty and hardship; some marriage promotion supporters advocate promoting marriage by excluding single-parent families from some public benefits. Marriage promotion teaches women to be dependent on a spouse instead of being economically independent. One randomized controlled study reported that the most effective marriage promotion program provided assistance for job stability. Long, George. "Lex Papia Poppaea". A Dictionary of Greek and Roman Antiquities: 691–692. LGBT parenting

Marc Ecko's Getting Up: Contents Under Pressure is a video game released in February 2006 for PlayStation 2, Windows. It was published by Atari, Inc. under license by Ecko Unlimited. There were two editions of the game, one being a limited edition and the other being the normal release; the game was re-published by Devolver Digital in December 2013 on Steam. The game focuses on an amateur graffiti artist known as Trane who uses graffiti and tagging as a way to protest against the corrupt dystopic city of New Radius, in a future world where freedom of expression is suppressed by a tyrannical, Orwellian city government; the gameplay is set up like a non-linear fighting game. The musical score for the game was produced by hip hop artist RJD2; the story begins in the New Radius Slums, where Trane, against the will of his grandmother, runs away from home to establish himself as a Graffiti artist. His first goal was to set himself as a notorious graffiti artist within the New Radius slums, his exploits reveals the state of New Radius, as a result of Sung's attempts at Gentrification of the city to make it look prosperous.

He has oppressed the lower class societies by eliminating the budget of liberal arts for more "prosperous" enterprises in the city, using the C. C. K to keep the slums and the city's seedy reputation suppressed through violence. Trane soon butts heads with the Vandals of New Radius led by Gabe who paints over his works. Realizing he needs to make his crew well known, he tags the city monorail which allows him to show his graffiti around New Radius in the face of both the C. C. K and the VaNR and defeating Gabe in a Graffiti battle which forces an alliance between both Crews. Soon enough, Trane is incapacitated by Decoy, who tells him about his father's past and how it all connects to Sung's rise to power, he tells Trane that the anti-graffiti campaign was a smokescreen to prevent artists such as Decoy from revealing the truth: Sung paid Trane's father to assassinate a rival candidate. Decoy had been tagging posters with the phrase "9/06", the day that Sung ordered the murder of Trane's father to cover up his involvement.

Taking his revolution to the next step, Trane begins tagging upper New Radius, the pristine part of the city which Sung upheld as the bastion of progress. Amongst the tagging campaign, Gabe betrays Trane under the threat of being killed by Shanna, an assassin in the employ of the New Radius News who uses recent events to make interesting news; this leads to Lower New Radius being attacked by C. C. K death squads, Decoy's death at the hands of New Radius News's assassin and lover Shanna. Swearing revenge, he plans a smear campaign against Sung by planting flyers which incriminate Sung for his role in the murder of another candidate. Sung's second in commnad J-Twizz joins up with Trane. From this point Trane can unleash the "Tag'n' Bag" attack when his hype meter fills by pressing the X button. Trane defeats Shanna. While the fight to overthrow Sung has ended, the fight for freedom is never finished for Trane as he continues his tagging campaigns to keep the city of New Radius always in question.

Marc Ecko has described the challenges of developing the game in interviews. These range from a missed Black Friday release date by saying "The code just wasn't ready" to communicating his vision to the developers: The gaming community has a natural tendency to take anything cool and make it cartoonish; that was a big learning curve. and the refusal of classification of the game in one market: I think it's demonization of graffiti, demonization of technology, the generational disconnect. I think. A mobile phone version of the game was announced in February 2005. On December 12, 2013 Devolver Digital re-released Getting Up: Contents Under Pressure onto Steam after acquiring the rights from the Atari bankruptcy that year. Marc Ecko's Getting Up: Contents Under Pressure received average to positive reviews, scoring 69 out of 100 in Metacritic's average, it received positive press from a few outlets, scoring 87% in GamesMaster magazine, 8.7 out of 10 on GameSpot. Common criticisms were related to the game's presentation and camera, although many outlets were impressed by the potential of the concept, hopes for a more refined sequel.

Edge magazine gave that title a 4/10 score and thought it ironic that the game was unpolished, given that it is "based on a culture of reputation and leaving a mark". Official US PlayStation Magazine was disappointed that the game was "so damn serious" in comparison to Jet Set Radio, a title with a similar premise. Penny Arcade criticized the game, calling it "God's punishment for an evil world."In an interview in Metro New York, Ecko was outspoken in his response to these critics, describing gamers as "the guys who got wedgies in high school" and "divas" with a "predisposition to have a bug up their ass for anything urban", who dismissed the game as riding on the coat-tails of Grand Theft Auto for having a black character on the cover. The reviewers, he says, are "slaves to the code" and not "slaves to the branding, products, or experience" as he would prefer, they unfairly compare the game to better-received titles, such as Prince of Persia: The Sands of Time, he admits that the game was "not as precise as I would have liked to s

Ireland on Sunday was a national Sunday newspaper published in Ireland from September 1997 until September 2006, when it was renamed the Irish Mail on Sunday. The newspaper was founded in 1996 as a sports-only newspaper called The Title, but was soon expanded into a general broadsheet Sunday newspaper with its founder, former County Meath Gaelic football player Liam Hayes, carrying on as editor; the paper was considered a'middle-market' publication. The Title was founded in 1996 by journalists Liam Cathal Dervan. Hayes, a former captain of the Meath Gaelic football team who made five All-Ireland final appearances, had worked with Dervan at the Meath Chronicle before spending seven years at the Sunday Press, a national newspaper which folded in 1995; the Title focused on sports, covering events from local to international in scale. The newspaper attracted a niche audience, garnering a circulation of 29,000. On 21 September 1997, the publication was relaunched as Ireland on Sunday, a three-section, full-colour broadsheet newspaper incorporating The Title as its tabloid-format sports section.

The venture was backed by a consortium of Irish and Irish-American investors, headed by the property developer Paschal Taggart. The consortium invested a total of £4m, with £1.5m being spent on aggressive marketing in the first year. The paper sought to capture some of the former readers of the Sunday Press, which had a circulation of 150,000 before its demise. Hayes said before launch that Ireland on Sunday aimed to "reflect the thoughts, the values and the desires of a 32-county Ireland"; the newspaper met with success and the first issue's run of 120,000 sold out. Circulation settling at 66,863 by March 1998, well above its launch-day estimate of 40,000 within two years. However, financial difficulties forced Hayes to remortgage his house and ask the company's directors for €50,800 each to keep the title afloat. In July 2000, Ireland on Sunday was sold to Scottish Radio Holdings for €10.16 million, with Hayes remaining in place as editor. However, in December of that year, he resigned from his post, to be replaced by former Evening Herald editor Paul Drury.

In July 2001 it was announced that Associated Newspapers, the national newspaper arm of the Daily Mail and General Trust, had bought Ireland on Sunday from Scottish Radio Holdings for £7.4 million. The circulation of the newspaper from January to June 2001 was 53,051; the Daily Mail publisher embarked on a campaign to reposition the middle-market title, slashing its cover price to 50c and bringing its design in line with the British Mail on Sunday. Its editor during this period, former Scotsman editor Martin Clarke, was accused of anglicising the Irish nationalist title. On 21 September 2006, DMGT announced that the previous weekend's edition of Ireland on Sunday had been the last under that masthead, completing its transition to becoming the Irish Mail on Sunday; the newspaper was informally considered to be the Irish edition of the Mail on Sunday and used a variation of that newspaper's masthead and editorial style, but DMGT's move made that transition complete. The UK edition of the Mail on Sunday was withdrawn from the Irish market in line with this.

IoS was aimed at the traditional Irish nationalist readers who had read The Sunday Press, leading it to adopt an Irish nationalist slant. It sold more copies in Northern Ireland per week than any other Dublin-based weekly newspaper. Founding editor Liam Hayes said that its tone was intended to be "humorous and responsible"; the paper's columnists included nationalist historian Tim Pat Coogan and publisher of the New York-based Irish Voice Niall O'Dowd. It attracted controversy on several occasions, clashing with Taoiseach Bertie Ahern over invasion of privacy and supporting swimmer Michelle Smith long after other newspapers had turned against her over allegations of doping

Ferdinand II was King of León and Galicia from 1157 to his death. Born in Toledo, Castile, he was the son of King Alfonso VII of León and Castile and of Berenguela, of the House of Barcelona. At his father's death, he received León and Galicia, while his brother Sancho received Castile and Toledo. Ferdinand earned the reputation of a good knight and hard fighter, but did not display political or organising faculty, he spent most of his first year as king in a dispute with his powerful nobles and an invasion by his brother Sancho III. In 1158 the two brothers met at Sahagun, peacefully solved the heritage matters. However, Sancho died in the same year, being succeeded by his child son Alfonso VIII, while Ferdinand occupied parts of Castile; the boundary troubles with Castile restarted in 1164: he met at Soria with the Lara family, who represented Alfonso VIII, a truce was established, allowing him to move against the Muslim Almoravids who still held much of southern Spain, to capture the cities of Alcántara and Alburquerque.

In the same year, Ferdinand defeated King Afonso I of Portugal, who, in 1163, had occupied Salamanca in retaliation for the repopulation of the area ordered by the King of León. In 1165 he married daughter of Afonso of Portugal. However, strife with Portugal was not put to an end by this move. In 1168 Afonso again felt menaced by Ferdinand II's repopulation of the area of Ciudad Rodrigo: he attacked Galicia, occupying Tui and the territory of Xinzo de Limia, former fiefs of his mother. However, as his troops were besieging the Muslim citadel of Badajoz, Ferdinand II was able to push the Portuguese out of Galicia and to rush to Badajoz; when Afonso saw the Leonese arrive he tried to flee, but he was disabled by a broken leg caused by a fall from his horse, made prisoner at one the city's gates. Afonso was obliged to surrender as his ransom all the conquests he had made in Galicia in the previous year. In the peace signed at Pontevedra the following year, Ferdinand got back twenty five castles, the cities of Cáceres, Trujillo, Santa Cruz and Montánchez lost by León.

When in the same years the Almoravids laid siege to the Portuguese city of Santarém, Ferdinand II came to help his father-in-law, helped to free the city from the menace. In 1170, Ferdinand created the military-religious Order of Santiago de Compostela, with the task to protect the city of Cáceres. Like the Order of Alcántara, it began as a knightly confraternity and took the name "Santiago" after St. James the apostle. In 1175 Pope Alexander III annulled Ferdinand II and Urraca of Portugal's marriage due to consanguinuity; the King remarried to Teresa Fernández de Traba, daughter of count Fernando Pérez de Traba, widow of count Nuño Pérez de Lara. In 1178 war against Castile broke out. Ferdinand surprised his nephew Alfonso VIII, occupied Castrojeriz and Dueñas, both lands of Teresa's first husband; the war was settled in 1180 with the peace of Tordesillas. In the same year his wife Teresa died. In 1184, after a series of failed attempts, the Almohad caliph Abu Yaqub Yusuf invaded Portugal with an army recruited in Northern Africa and, in May, besieged Afonso I in Santarém.

In 1185 Ferdinand married for the third time to Urraca López de Haro, his mistress since 1180. Urraca tried in vain to have Alfonso IX, first son of Ferdinand II, declared illegitimate, to favour her son Sancho. Ferdinand II died in 1188 at Benavente, while returning from a pilgrimage to Santiago de Compostela, he was buried in the cathedral of Compostela. In 1230 Forty two years after Ferdinand II's death his namesake grandson Ferdinand III of Castile united Castile with Leon permanently. Ferdinand married Urraca of Portugal around 1165, they had one son: Alfonso IX. Following her repudiation, he formed a relationship with Teresa Fernández de Traba, daughter of count Fernando Pérez de Traba, in August 1179 he married her, having: Ferdinand, legitimized through his parents' subsequent marriage child, b. and d. 6 February 1180, whose birth led to the death of its motherHe formed a liaison with Urraca López de Haro, daughter of Lope Díaz I de Haro, whom he married in May 1187, having: García Alfonso, b.1184, legitimized through the subsequent marriage of his parents, died before his father.

Sancho, lord of Fines Busk, M. M; the history of Spain and Portugal from B. C. 1000 to A. D. 1814, Baldwin and Cradock, 1833. Leese, Thelma Anna, Blood royal: issue of the kings and queens of medieval England, 1066–1399, Heritage Books, 1996. Medieval Iberia: an encyclopedia, Ed. E. Michael Gerlis and Samuel G. Armistead, Taylor & Francis, 2003. Morton, Nicholas; the Medieval Military Orders: 1120-1314. Routledge. ISBN 978-1-31786-147-8. Szabolcs de Vajay, "From Alfonso VIII to Alfonso X" in Studies in Genealogy and Family History in Tribute to Charles Evans on the Occasion of his Eightieth Birthday, 1989, pp. 366–417. Cawley, Fernando II, king of León 1157–1188, Medieval Lands database, Foundation for Medieval Genealogy Cawley, Medieval Lands Project on the kings and counts of Castile & León, Medieval Lands database, Foundation for Medieval Genealogy