Fermat's Last Theorem

In number theory, Fermat's Last Theorem states that no three positive integers a, b, c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have an infinite number of solutions; the proposition was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica. However, there were doubts that he had a correct proof because his claim was published by his son without his consent and after his death. After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles, formally published in 1995, it proved much of the modularity theorem and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century, it is among the most notable theorems in the history of mathematics and prior to its proof was in the Guinness Book of World Records as the "most difficult mathematical problem" in part because the theorem has the largest number of unsuccessful proofs.

The Pythagorean equation, x2 + y2 = z2, has an infinite number of positive integer solutions for x, y, z. Around 1637, Fermat wrote in the margin of a book that the more general equation an + bn = cn had no solutions in positive integers if n is an integer greater than 2. Although he claimed to have a general proof of his conjecture, Fermat left no details of his proof, no proof by him has been found, his claim was discovered some 30 years after his death. This claim, which came to be known as Fermat's Last Theorem, stood unsolved for the next three and a half centuries; the claim became one of the most notable unsolved problems of mathematics. Attempts to prove it prompted substantial development in number theory, over time Fermat's Last Theorem gained prominence as an unsolved problem in mathematics; the special case n = 4, proved by Fermat himself, is sufficient to establish that if the theorem is false for some exponent n, not a prime number, it must be false for some smaller n, so only prime values of n need further investigation.

Over the next two centuries, the conjecture was proved for only the primes 3, 5, 7, although Sophie Germain innovated and proved an approach, relevant to an entire class of primes. In the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes, leaving irregular primes to be analyzed individually. Building on Kummer's work and using sophisticated computer studies, other mathematicians were able to extend the proof to cover all prime exponents up to four million, but a proof for all exponents was inaccessible. Separately, around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two different areas of mathematics. Known at the time as the Taniyama–Shimura conjecture, it stood on its own, with no apparent connection to Fermat's Last Theorem, it was seen as significant and important in its own right, but was considered inaccessible to proof. In 1984, Gerhard Frey noticed an apparent link between these two unrelated and unsolved problems.

An outline suggesting this could be proved was given by Frey. The full proof that the two problems were linked was accomplished in 1986 by Ken Ribet, building on a partial proof by Jean-Pierre Serre, who proved all but one part known as the "epsilon conjecture"; these papers by Frey and Ribet showed that if the Taniyama–Shimura conjecture could be proven for at least the semi-stable class of elliptic curves, a proof of Fermat's Last Theorem would follow automatically. The connection is described below: any solution that could contradict Fermat's Last Theorem could be used to contradict the Taniyama–Shimura conjecture. So if the modularity theorem were found to be true by definition no solution contradicting Fermat's Last Theorem could exist, which would therefore have to be true as well. Although both problems were daunting and considered to be "completely inaccessible" to proof at the time, this was the first suggestion of a route by which Fermat's Last Theorem could be extended and proved for all numbers, not just some numbers.

Unlike Fermat's Last Theorem, the Taniyama–Shimura conjecture was a major active research area and viewed as more within reach of contemporary mathematics. However, general opinion was that this showed the impracticality of proving the Taniyama–Shimura conjecture. Mathematician John Coates' quoted reaction was a common one: "I myself was sceptical that the beautiful link between Fermat’s Last Theorem and the Taniyama–Shimura conjecture would lead to anything, because I must confess I did not think that the Taniyama–Shimura conjecture was accessible to proof. Beautiful though this problem was, it seemed impossible to prove. I must confess I thought I wouldn’t see it proved in my lifetime."On hearing that Ribet had proven Frey's link to be correct, English mathematician Andrew Wiles, who had a childhood fascination with Fermat's Last Theorem and ha

The Children of the Night (album)

The Children of the Night is the third studio album by Swedish death metal band Tribulation. It was released on 20 April 2015 through Century Media Records; the album's sound is a stark departure from the death metal style of Tribulation's previous records. The Children of the Night features a heavy metal and extreme metal sound with elements from psychedelic rock, progressive metal, thrash metal, classic rock, hard rock, gothic rock, it attributes influences to various rock and metal music acts such as Mercyful Fate, At the Gates, Led Zeppelin, the Doors, Iron Maiden, Deep Purple and Pink Floyd. The title of the album is a reference to Kiss's 1982 album Creatures of the Night; the album received positive reviews from music critics. Chris Dick of Decibel magazine wrote: "There are few bands who are capable of Children of the Night". Pitchfork critic Grayson Haver Currin described the album as "a heavy metal record that wanders beyond any comfort zone" and "a sprawling, compulsory tale that doesn’t turn dull".

Loudwire's Joe DiVita thought that the record "offers a refreshing take on a beloved style with enough extreme metal elements in tact that should please fans on both sides of the fence." Michael Nelson of Stereogum regarded it as "the best record he has heard in 2015."Pitchfork's Brandon Stosuy listed The Children of the Night as number two on his list of "The Best Metal Albums of 2015". Spin magazine critic Colin Joyce listed the album as number six on the publication's list of "The 20 Best Metal Albums of 2015". "Strange Gateways Beckon" – 4:29 "Melancholia" – 5:17 "In the Dreams of the Dead" – 5:52 "Winds" – 6:52 "Själaflykt" – 5:52 "The Motherhood of God" – 5:23 "Strains of Horror" – 6:14 "Holy Libations" – 6:34 "Cauda Pavonis" – 2:55 "Music from the Other" – 7:04 Album personnel as adapted from album liner notes. TribulationJohannes Andersson – vocals, bass guitar, backing vocals Adam Zaars – guitar, backing vocals, xylophone, layout Jonathan Hultén – guitar, backing vocals, cover art Jakob Ljungbergdrums, percussionOther personnelMartin Borgh – additional instruments Ola Ersfjord – producer, recording engineer, mixing Chris Commonmastering Johan Voxberg – drum technician Susanna Berglund – photography Linda Åkerberg – photography Official website

Grama Vikas Kendra

The Nalpathimala Grama Vikas Kendra is the extension centre of the Mahatma Gandhi University, in India, at its main campus in Kottayam, Kerala. The major activities of the Grama Vikas Kendra are centered on the village of Nalpathimala; the Kendra was commissioned by the Vice Chancellor, Dr. U. R. Ananthamurthy, with a vision ‘to add a chethana dimension to the chinthana preoccupations of the University’. Being an extension centre of the Mahatma Gandhi University, the Kendra put forth a paradigm for achieving Gramaswaraj through linking the campus with the community; the Kendra is run by the Department of Adult Education and Field Outreach in association with National Service Scheme. Dr. C. Thomas Abraham was instrumental in forming the idea of Grama Vikas Kendra, shaping and transforming it as replicable model for campus-community partnership; the Home-Stay Camp is the major element in the Campus-Community Partnership programme. As per this programme, the students and teachers spend a few days with the village community.

During the camp days, these students and teachers stay with the families in the village. The Home-Stay Camp has three major aspects; this is a reciprocal programme between the community. Both the campus and the community can share a lot of valuable information between each other. Through this programme, the community gets informative seminars and training programmes on various subjects of the time; the continuing workshops on Panchayathi Raj, the environment, women empowerment are the notables among these programmes. Mahatma Gandhi, in his Constructive Programme, put first priority to the communal unity. For him it was an unbreakable heart unity; each family in the village accommodates the campers from different a caste and social status. It is helpful to eliminate prejudices about the ‘other’ and ensure national unity. In this way, the Home-Stay camps fulfill the grand vision of the Mahatma. Moreover through the Home-Stay camps, the academic community can understand the needs of society, so that the research programmes can be reoriented in accordance with the expectations of society.

Home-Stay camps create a feeling that the academic community has a constructive role to play in the development of the society. Therefore, manual labour is an important programme in its schedule; the Kendra has organized 100 Home-Stay camps whereby more than a thousand volunteers across the country have participated in it. In short, the Home-Stay camps could impart its participants a new vision about village and education that are not handled by the classroom syllabus; the Kendra is undertaking the following programme for women empowerment: seminars on the rights of the women. There are 100 SHGs affiliated with the Kendra; the message of swadeshi is promoted through these SHGs. The major employment programmes of these SHGs are for work in the production of paper-bags, herbal medicines, vermi-compost and curry powder; the Kendra plays a major role in the development of the area of its functioning with the help of various governmental and non-governmental agencies, such as Government Poly-technique, NABARD, CWRDM, Department of Soil Conservation, Nirmal-2000, OMMI-HRD project and CHASS.

Through the job-training programmes, the unemployment rate in the area could be reduced to a great extent. The Kendra work out construction of sanitary latrines, maintenance of houses, wells in the village of Nalpathimala. In order to solve the water scarcity problem in the village, the Kendra had chalked out a "water literacy campaign" that could improve awareness about the importance of ferro-cement water tanks, percolation pits and other methods in conserving water; the Kendra is organizing various programmes for the senior citizens through medical camps, recreation programmes, etc. The frequent meetings of the Children' Forums is intended to enrich their leadership quality and creativity; the Kendra provides a remedial education programme for the educationally-backward students to improve their standard. The Kendra is bringing out a newsletter. There are various courses such as yoga, geriatric social work, other capacity building programmes held in the Kendra; the Kendra is conducting a training programme in community organization.

A person, interested in participatory rural development can apply for the training programme. Selected candidates are appointed as a Grama Vikas Volunteer, they will be assigned with the charge of various activities at the Grama Vikas Kendra. In addition to that, the Kendra facilitates the GVVs to attend the training programmes organized by the governmental and non-governmental agencies, with regard to rural development