Borås is a city and the seat of Borås Municipality, Västra Götaland County, Sweden. It had 66,273 inhabitants in 2010. Borås is located at the point of two crossing railways, among them the railway between Gothenburg and Kalmar, is considered the Swedish city gaining the most from the nationwide railway system laid between 1870 and 1910; the city of Borås received its privileges in 1621 by King Gustav II Adolf. The reason was to give local pedlars a legal place for vending their merchandise; the city developed. After a century it had increased to over 2,000 inhabitants. Borås has been ravaged by fires four times: in 1681, 1727, 1822 and 1827; the Caroli church is the oldest of Borås's buildings, has withstood all fires. In its 2017 report, Police in Sweden placed the Norrby, Hässleholmen and Hulta districts in the most severe category of urban areas with high crime rates; the city arms depicts two pairs of sheep shears, a tribute to the vast number of smiths in the town in early history. It holds the Swedish record in the number of established mail-order firms.

The company Swedac is based in the city, as well as Ericsson, which has a large manufacturing plant where Mini-Link microwave radios are manufactured. Worldwide clothing retailer H&M have their worldwide Online office based in the city as well. Outside the city in the industrial park of Viared, there are many companies specializing in logistics. Industries in Borås have close collaboration with the University College of Borås as well as the SP Technical Research Institute of Sweden, the largest technical research institute of Sweden, both located in Borås. Besides being the home of the official Swedish Kilogram-weight and the atomic clock that sets the national time, SP conducts various testing and research to promote consumer safety. Borås is a railway junction; the Coast-to-Coast Line from Gothenburg to Kalmar runs through the city centre. At the Borås Railway Station it meets the Viskadal Line. Continuous passenger trains run through the city along the two latter lines; as of 2019 SJ runs 3-4 daily trains in each direction between Kalmar.

The regional traffic agency Västtrafik runs frequent trains between Borås and Gothenburg and Varberg. All passenger trains stop at Borås railway station. Trains on the northern line to and from Herrljunga stop at Knalleland halt within the city; the most successful sports team is the football team IF Elfsborg, six times Swedish champions and three times winner of the Swedish cup. Elfsborg was founded in 1904, but received its current name in 1906; the team's home arena, since 2005, is Borås Arena in Borås, with a capacity of 16000. Elfsborg play in black. Other teams are Borås Hockey Club, M7 and Borås Rhinos, an American football team; the table tennis club, Mariedals IK has three players in the Swedish youth team: Hampus Soderlund, Mattias Översjö and Jimmy Ojakangas. Other sports that can be practiced in Borås are: archery and field, handball, table tennis, skiing, sport shooting, golf, bowling and various martial arts. Borås Judoclub has tried to create a national center for judo. Members from the club that have participated in the olympics include Lars Adolfsson.

Boråshallen is the largest indoor sports hall in Borås, where the Mariedal cup, an indoor football competition, is held annually in October or November. The competition started in 1978 in its current form. Borås hosted the water polo events of the XIII FINA World Masters Championships 2010, from July 28 to August 6; the city hosted the 2019 European Athletics U20 Championships. Other sports clubs located in Borås include: Bergdalens IK Borås AIK Hemvärnets Musikkår Borås – a marching and sitting orchestra Borås kulturskola – School of the Arts. Borås kulturhus – theater, museum of art Borås Tidning – the local newspaper. Textilmuseet – a museum about textile manufacturing and clothes. Borås djurpark – The local zoo, they have raised a number of different animals and exported lions to other zoos. Åhaga – A renovated building used to maintain and repair locomotives. Now used for conferences and concerts; every summer, a number of free outdoor concerts are held on the town square on Thursday evenings.

The city has been recognized as a city of outdoor sculptures, such as Catafalque by Sean Henry, Carl Fredrik Reuterswärd's knotted non-violence pistol and a huge bronze version of Pinocchio called Walking to Borås by the American pop artist Jim Dine. During the holiday All Saints' Day, there is an annual gaming convention called Borås Spelkonvent, where people youths, meet to play various board games, Warhammer 40k and Magic: The Gathering among other games. Borås has a cold oceanic climate, influenced by Atlantic trade winds travelling without major obstacles over the land, with low mean temperatures, compared to coastal areas and the more urban metropolis, near to the west. Considering it being inland and quite far south, summer temperatures are rather subdued by Swedish standards, whilst winter average highs hover just above freezing. Being inland still enables severe extremes to sometimes occur, such as a 36 °C reading from July 1901, the warmest July temperature on record in Sweden, although not a n

Lumut known as Kampong Lumut, is a settlement area in Belait District, Brunei. It is the common name given to a group of village subdivisions and residential communities in the subdistrict of Liang. Archaeological evidence found in Lumut suggests that human activity had existed in the area as early as the fourteenth century. However, the characteristics of the evidence suggests that the site was for burial rather than residential purpose; the area which constitutes Lumut comprises several village subdivisions, the third- and lowest-level administrative division of Brunei. The villages in Lumut are subdivisions of a subdistrict of Belait; each village may correspond to a postcode area. These subdivisions in Lumut as designated by the Survey Department include: Kampong Lumut Kampong Lumut Tersusun Kampong Sungai Bakong Kampong Sungai Kuru Kampong Sungai Tali Kampong Sungai Taring Perkhemahan Lumut Perumahan Negara Lumut Kawasan 1 Perumahan Negara Lumut Kawasan 2 Skim Tanah Kurnia Rakyat Jati LumutHowever, the Belait District Office, which de facto administers the communities in the district, governs the population in Lumut as two village communities, namely Kampong Lumut I and II, whereby each has its own village head.

The School of Science and Engineering campus of Brunei Polytechnic is located in Lumut, which offers post-secondary, vocational courses in engineering field, taking advantage of the proximity to the oil and gas industries in the district. There is one secondary school in Lumut: Pengiran Anak Puteri Hajah Rashidah Sa’adatul Bolkiah Secondary School provides five years of secondary education leading up to'O' Level qualification. Primary education is available in Lumut, provided in two primary schools, namely Lumut Primary School and Sungai Tali Primary School. There are three primary religious schools in Lumut, namely Lumut Religious School, Sungai Tali Religious School and Sungai Taring Religious School, they provide primary religious education, compulsory for Muslim children in the country. Economy in Lumut is based on the petroleum industry, it is the location of the Brunei LNG plant, a 130-hectare liquified natural gas plant with annual production capacity of 6.7 million tonnes. Lumut has a small retail economy.

There are a few commercial buildings and several shops grocery stores and restaurants, for the consumption of the residents. Lumut is served by Jalan Seria–Kuala Belait, a two-lane road which traverses the area. Lumut Bypass is a highway. There is a third access but it is not open to public as it traverses Lumut Camp, a military area. Lumut is one of the locations for a government housing scheme. Zainab Mosque serves the need of the Muslim residents for Islamic congregational prayers and activities, in particular the Jumu'ah or Friday prayers; the mosque was inaugurated on 22 May 1998 by the Sultan of Brunei. It can accommodate 2,000 worshippers at one time. Lumut Post Office is the sole post office in the area. Lumut is a coastal settlement and Lumut Beach stretches along the coastline where the area meets the South China Sea. A section of the beach has been made a recreation park for the public. Liang Lumut Recreation Club is a venue for recreational events, it has a multipurpose hall as well as a small waterpark

Fermat's right triangle theorem is a non-existence proof in number theory, the only complete proof left by Pierre de Fermat. It has several equivalent formulations: If three square numbers form an arithmetic progression the gap between consecutive numbers in the progression cannot itself be square. There do not exist two Pythagorean triangles in which the two legs of one triangle are the leg and hypotenuse of the other triangle. A right triangle for which all three side lengths are rational numbers cannot have an area, the square of a rational number. An area defined in this way is called a congruent number, so no congruent number can be square. A right triangle and a square with equal areas cannot have all sides commensurate with each other; the only rational points on the elliptic curve y 2 = x are and. The Diophantine equation x 4 − y 4 = z 2 has no integer solution. An immediate consequence of the last of these formulations is that Fermat's last theorem is true for the exponent n = 4, hence for any multiple of 4.

In 1225, Fibonacci was challenged to find a construction for triples of square numbers that are spaced from each other, forming an arithmetic progression, for the spacing between these numbers, which he called a congruum. One way of describing Fibonacci's solution is that the numbers to be squared are the difference of legs and sum of legs of a Pythagorean triangle, that the congruum is four times the area of the same triangle. In his work on the congruum problem, published in The Book of Squares, Fibonacci observed that it is impossible for a congruum to be a square number itself, but did not present a satisfactory proof of this fact. If three squares a 2, b 2, c 2 could form an arithmetic progression whose congruum was a square d 2 these numbers would satisfy the Diophantine equations a 2 + d 2 = b 2 and b 2 + d 2 = c 2; that is, by the Pythagorean theorem, they would form two integer-sided right triangles in which the pair gives one leg and the hypotenuse of the smaller triangle and the same pair forms the two legs of the larger triangle.

But if no square congruum can exist there can be no two integer right triangles that share two sides in this way. Because the congrua are the numbers that are four times the area of a Pythagorean triangle, multiplication by four does not change whether a number is square, the existence of a square congruum is equivalent to the existence of a Pythagorean triangle with a square area, it is this variant of the problem that Fermat's proof concerns: he shows that there is no such triangle. In considering this problem, Fermat was inspired not by Fibonacci but by an edition of Diophantus published by Claude Gaspard Bachet de Méziriac; this book described various special right triangles whose areas had forms related to squares, but did not consider the case of areas that were themselves square. By rearranging the equations for the two Pythagorean triangles above, multiplying them together, one obtains the single Diophantine equation b 4 − d 4 = = a 2 c 2 which can be simplified to b 4 − d 4 = e 2. Conversely, any solution to this equation could be factored to give a square congruum.

Thus, the solvability of this equation is equivalent to the existence of a square congruum. But, if Fermat's last theorem were false for the exponent n = 4 squaring one of the three numbers in any counterexam