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HISTORY OF MATHEMATICS – LECTURE 16 – WEDNESDAY 3 RD NOVEMBER Carl Friedrich Gauss Known as the “Prince of Mathematics“, Carl Friedrich Gauss (1777 1855) is universally regarded as one of the greatest mathematicians of all time. Although hailing from a poor family in Brunswick, Germany, the talent of Gauss was obvious from a young age, and he regularly astonished his school masters with his knowledge. His extraordinary ability as a child caused the Duke of Brunswick to finance Gauss through preparatory school, and then the University of Göttingen from 1795 1798. Whilst there Gauss made a series of interesting discoveries, generalizing the polygons which can be constructed using straight edge and compass (an ancient problem that was thought to have been solved), and proving the quadratic reciprocity law while still only 18. Later that year he proved that every positive integer can be written as the sum of three triangular numbers, and gave a version of the prime number theorem, which we will consider below. He also invented modular arithmetic, which led to the publication in 1801 of his most famous work, Disquisitiones Arithmetica (Arithmetic Investigations). Prime Number Theorem We proved in Lecture 5 that there are infinitely many prime numbers. So since we cannot determine all of them, mathematicians have long been interested in how they are distributed amongst the natural numbers. No exact answer has ever been found, and so instead we try to approximate how many primes exist between 1 and any given number. We (confusingly) use π to denote this, and so π (10) = 4, since there are 4 prime numbers less than 10. Given that there are infinitely many prime numbers, π (x) must tend to infinity as x does, but the number of primes does tend to decrease as x gets larger (see below). x π (x) Number of new primes 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Gauss formed this table when was 15, and conjectured that the number of primes less than or equal to the natural number x is given approximately by ׬ ୪୭୥ ௡ ݀݊ . While not a perfect approximation, it is very accurate (for example when x = 3,000,000,
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lecture16 -...

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