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The Beatles

The Beatles were an English rock band formed in Liverpool in 1960. With a line-up comprising John Lennon, Paul McCartney, George Harrison and Ringo Starr, they are regarded as the most influential band of all time; the group were integral to the development of 1960s counterculture and popular music's recognition as an art form. Rooted in skiffle, beat and 1950s rock and roll, their sound incorporated elements of classical music and traditional pop in innovative ways; as pioneers in recording and artistic presentation, the group revolutionised many aspects of the music industry and were publicised as leaders of the era's youth and sociocultural movements. Led by primary songwriters Lennon and McCartney, the Beatles built their reputation playing clubs in Liverpool and Hamburg over three years from 1960 with Stuart Sutcliffe playing bass; the core trio of Lennon, McCartney and Harrison, together since 1958, went through a succession of drummers, including Pete Best, before asking Starr to join them in 1962.

Manager Brian Epstein moulded them into a professional act and producer George Martin guided and developed their recordings expanding their domestic success after their first hit, "Love Me Do", in late 1962. As their popularity grew into the intense fan frenzy dubbed "Beatlemania", the band acquired the nickname "the Fab Four", with Epstein and other members of the band's entourage sometimes given the informal title of "fifth Beatle". By early 1964, the Beatles were international stars, leading the "British Invasion" of the United States pop market and breaking numerous sales records, they soon made their film debut with A Hard Day's Night. From 1965 onwards, they produced innovative recordings, including the albums Rubber Soul and Sgt. Pepper's Lonely Hearts Club Band, enjoyed further commercial success with The Beatles and Abbey Road. In 1968, they founded Apple Corps, a multi-armed multimedia corporation that continues to oversee projects related to the band's legacy. After the group's break-up in 1970, all four members enjoyed success as solo artists.

Lennon was shot and killed in December 1980, Harrison died of lung cancer in November 2001. McCartney and Starr remain musically active; the Beatles are the best-selling music act of all time, with certified sales of over 183 million units in the US and estimated sales of 600 million units worldwide. They hold the record for most number-one albums on the UK Albums Chart, most number-one hits on the Billboard Hot 100 chart, most singles sold in the UK; the group were inducted into the Rock and Roll Hall of Fame in 1988, all four main members were inducted individually between 1994 and 2015. In 2008, the group topped Billboard's list of the all-time most successful artists on the Billboard Hot 100; the band have received seven Grammy Awards, four Brit Awards, an Academy Award and fifteen Ivor Novello Awards. Time named them among the 20th century's 100 most important people. In March 1957, John Lennon aged sixteen, formed a skiffle group with several friends from Quarry Bank High School in Liverpool.

They called themselves the Blackjacks, before changing their name to the Quarrymen after discovering that another local group was using the name. Fifteen-year-old Paul McCartney joined them as a rhythm guitarist shortly after he and Lennon met that July. In February 1958, McCartney invited his friend George Harrison to watch the band; the fifteen-year-old auditioned for Lennon, impressing him with his playing, but Lennon thought Harrison was too young for the band. After a month of Harrison's persistence, during a second meeting, he performed the lead guitar part of the instrumental song "Raunchy" on the upper deck of a Liverpool bus, they enlisted him as their lead guitarist. By January 1959, Lennon's Quarry Bank friends had left the group, he began his studies at the Liverpool College of Art; the three guitarists, billing themselves as Johnny and the Moondogs, were playing rock and roll whenever they could find a drummer. Lennon's art school friend Stuart Sutcliffe, who had just sold one of his paintings and was persuaded to purchase a bass guitar with the proceeds, joined in January 1960, it was he who suggested changing the band's name to Beatals, as a tribute to Buddy Holly and the Crickets.

They used this name until May, when they became the Silver Beetles, before undertaking a brief tour of Scotland as the backing group for pop singer and fellow Liverpudlian Johnny Gentle. By early July, they had refashioned themselves as the Silver Beatles, by the middle of August shortened the name to The Beatles. Allan Williams, the Beatles' unofficial manager, arranged a residency for them in Hamburg, but lacking a full-time drummer they auditioned and hired Pete Best in mid-August 1960; the band, now a five-piece, left four days contracted to club owner Bruno Koschmider for what would be a 3​1⁄2-month residency. Beatles historian Mark Lewisohn writes: "They pulled into Hamburg at dusk on 17 August, the time when the red-light area comes to life... flashing neon lights screamed out the various entertainment on offer, while scantily clad women sat unabashed in shop windows waiting for business opportunities." Koschmider had converted a couple of strip clubs in the district into music venues, he placed the Beatles at the Indra Club.

After closing Indra due to noise complaints, he moved them to the Kaiserkeller in October. When he learned they had been performing at the rival Top Ten Club in breach of their contract, he gave the band one mont

James Kondo

James Kondo is an executive whose career spans social, government and academic sectors. From January 2019, he is Chairman of International House of Japan. After graduating from Keio High School, Kondo studied at Brown University, graduated from Keio University with a BA in Economics and Harvard Business School with an MBA, he is a Yale World Fellow at Yale University. On January 1, 2019, Kondo was appointed as Chairman at International House of Japan after serving as a Trustee and a Director in previous years, he serves as a Global Trustee of Asia Society and Co-Chair of Asia Society Japan Center. He is a Kenjin-Tatsujin Member of nonprofit organization Ashinaga. In 2011, he co-founded Beyond Tomorrow, a global fund for educational assistance to support youth in need, offering scholarship programs focused on leadership training, it first began as an educational fund supporting young victims of 2011 Tohoku disaster. In 2007, Kondo co-founded a nonprofit organization TABLE FOR TWO International with fellow Young Global Leaders from World Economic Forum.

Kondo serves as Co-Chair of Silicon Valley Japan Platform which connects business leaders in Silicon Valley and Japan. He serves as Representative Director at World Economic Forum Centre for the Fourth Industrial Revolution Japan, one of four global centres set up in the world to help shape governance for new technologies that are transforming our world, he serves as Senior Advisor at Geodesic Capital. Kondo served as Chairman of Twitter Japan and Vice President of Twitter Inc. Kondo co-founded Asia Pacific Initiative in 2011, has been serving as a board member and President. In 2010, he joined the Government of Japan and served in various capacities as Counsellor at the Prime Minister's Office. From 2011 to 2012, he became a Special Adviser to the Cabinet Office, he was a management consultant at McKinsey & Company for fifteen years from 1990 where he served as a core member of the firm's global strategy group and its economic think tank, McKinsey Global Institute. He co-founded Health and Global Policy Institute in 2004 and served as Vice Chairman and President until 2009.

Kondo is a Visiting Professor at Hitotsubashi University Business School conducting Business Government and International Economy course since 2011. He joined MIT Media Lab as a Visiting Scientist from 2014 to 2017. In 2003, Kondo joined University of Tokyo where he co-established Healthcare and Social Policy Program and served as a Co-Director until 2009. New Asian Leader, World Economic Forum Young Global Leader, World Economic Forum Global Agenda Council member, World Economic Forum US-Japan Leadership Program Delegate, The United States-Japan Foundation Asia 21 Fellow, Asia Society International Fellow, Center for Strategic and International Studies Inamori Fellow, Inamori Foundation Richard von Weizsäcker Fellow, Robert Bosch Academy Kondo Masaakira James - Chairman, Board of Directors, International House of Japan James Kondo on Twitter James Kondo - President, Asia Pacific Initiative Biography - James Kondo, Hitotsubashi University Business School Asia 21 Fellows, Class of 2006 TABLE FOR TWO Beyond Tomorrow

Congruum

In number theory, a congruum is the difference between successive square numbers in an arithmetic progression of three squares. That is, if x2, y2, z2 are three square numbers that are spaced apart from each other the spacing between them, z2 − y2 = y2 − x2, is called a congruum; the congruum problem is the problem of finding squares in arithmetic progression and their associated congrua. It can be formalized as a Diophantine equation: find integers x, y, z such that y 2 − x 2 = z 2 − y 2; when this equation is satisfied, both sides of the equation equal the congruum. Fibonacci solved the congruum problem by finding a parameterized formula for generating all congrua, together with their associated arithmetic progressions. According to this formula, each congruum is four times the area of a Pythagorean triangle. Congrua are closely connected with congruent numbers: every congruum is a congruent number, every congruent number is a congruum multiplied by the square of a rational number; as an example, the number 96 is a congruum because it is the difference between adjacent squares in the sequence 4, 100, 196.

The first few congrua are: 24, 96, 120, 216, 240, 336, 384, 480, 600, 720 …. The congruum problem was posed in 1225, as part of a mathematical tournament held by Frederick II, Holy Roman Emperor, answered at that time by Fibonacci, who recorded his work on this problem in his Book of Squares. Fibonacci was aware that it is impossible for a congruum to itself be a square, but did not give a satisfactory proof of this fact. Geometrically, this means that it is not possible for the pair of legs of a Pythagorean triangle to be the leg and hypotenuse of another Pythagorean triangle. A proof was given by Pierre de Fermat, the result is now known as Fermat's right triangle theorem. Fermat conjectured, Leonhard Euler proved, that there is no sequence of four squares in arithmetic progression; the congruum problem may be solved by choosing two distinct positive integers m and n. The middle square of the associated arithmetic progression of squares is 2, the other two squares may be found by adding or subtracting the congruum.

Additionally, multiplying a congruum by a square number produces another congruum, whose progression of squares is multiplied by the same factor. All solutions arise in one of these two ways. For instance, the congruum 96 can be constructed by these formulas with m = 3 and n = 1, while the congruum 216 is obtained by multiplying the smaller congruum 24 by the square number 9. An equivalent formulation of this solution, given by Bernard Frénicle de Bessy, is that for the three squares in arithmetic progression x2, y2, z2, the middle number y is the hypotenuse of a Pythagorean triangle and the other two numbers x and z are the difference and sum of the triangle's two legs; the congruum itself is four times the area of the same Pythagorean triangle. The example of an arithmetic progression with the congruum 96 can be obtained in this way from a right triangle with side and hypotenuse lengths 6, 8, 10. A congruent number is defined as the area of a right triangle with rational sides; because every congruum can be obtained as the area of a Pythagorean triangle, it follows that every congruum is congruent.

Conversely, every congruent number is a congruum multiplied by the square of a rational number. However, testing whether a number is a congruum is much easier than testing whether a number is congruent. For the congruum problem, the parameterized solution reduces this testing problem to checking a finite set of parameter values. In contrast, for the congruent number problem, a finite testing procedure is known only conjecturally, via Tunnell's theorem, under the assumption that the Birch and Swinnerton-Dyer conjecture is true. Automedian triangle, a triangle for which the squares on the three sides form an arithmetic progression Spiral of Theodorus, formed by right triangles whose sides, when squared, form an infinite arithmetic progression Weisstein, Eric W. "Congruum Problem". MathWorld