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HISTORY OF MATHEMATICS – LECTURE 20 – WEDNESDAY 1 ST DECEMBER Solved and Unsolved Problems During this course we have looked at the development of mathematics, and how seemingly intractable problems were solved or overcome. In particular we have considered how zero was not initially recognized, that negative solutions to equations were regarded as “absurd”, irrational numbers were not thought to exist, and complex numbers were not valid. In the 20th century there were more people employed to do mathematics than in all of history to that point. This led to many problems being solved, but many others being created, some of which we still cannot solve. In this lecture we will consider five famous problems that were solved during the second half of the 20th century, and five others that we still do not know the answer to. Paul Erd ő s In terms of the number of research papers published, Paul Erd ő s (1913 1996) was the most prolific mathematician of all time, with an output of 1475 papers, written with 511 different collaborators. Born in Hungary, he lived in home country until the age of 21 before starting a journey that led to him having no permanent home and no possessions from 1971 onwards, generally just moving to the house of another mathematician, writing a paper, and then moving on. While regarded as a world class mathematician, gaining 15 honorary doctorates and membership of the Royal Society, Erd ő s was equally famous for his eccentric behavior, which has inspired three biographies and a movie. When arriving at the house of another mathematician he would announce “My brain is open!” and then work on problems, often for up to 20 hours a day, fueled largely by coffee and amphetamines. Money did not mean a lot to Erd ő s, with him once giving away all but $720 of a $50,000 prize that he won to students and those who could solve his challenge problems. However he did accept a $500 bet that he could not stop taking amphetamines for a month. Erd ő s won the bet, but then complained that mathematical progress had been set back by a month as a result! Prime Number Theorem (Solved) In Lecture 16 we looked at how Gauss conjectured the Prime Number Theorem, which we generally write today as π (x) x/log x. No proof was found until 1896, when (independently) Jacques Hadamard (1865 1963) and Charles Jean de la Vallée Poussin (1866 1962) came up with long and complicated demonstrations of why the result holds. In 1949 Erd ő s and Atle Selberg (1917 2007) derived a (more) elementary proof, which was regarded as a great triumph. There has been controversy ever since as to how much of the work was done by the two involved, but while Selberg was awarded the Fields Medal in 1950, Erd ő s was overlooked, and never won the award.
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Terrence Tao One of the world’s greatest living mathematicians, Terrence Tao (born 1975) is an Australian mathematician of Chinese descent working at UCLA. His ability in mathematics was obvious from a very
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This note was uploaded on 09/22/2011 for the course MAC 2311 taught by Professor Evinson during the Spring '08 term at University of Central Florida.

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lecture20 -...

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