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PublicLecture - Mathematics: Beauty and the Beast Walter...

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1 Mathematics: Beauty and the Beast Walter Tholen York University Toronto
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2 EXAMPLE 1: Prime Numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, … 40 = 2·20 = 2·2·10 = 2·2·2·5 2006 = 2·1003 = 2·17·59 The largest known prime number (as of December 2005) has 9,152,052 digits. (It’s the 43 rd Mersenne prime number.)
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3 Euclid (~300BC): There are infinitely many prime numbers . “Whenever you give me a finite list p 1 , p 2 , …. ., p n of n primes, then I can give you (in principle) another prime p that is not yet in the list.”
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4 N = p 1 · p 2 · …. · p n + 1 N has a prime factor p. That prime factor p cannot be one of p 1 , p 2 , …, p n , for if it were, p would not only be a divisor of N, but also of N – 1 = p 1 · p 2 · …. · p n : impossible!
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5 Twin primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, … Are there infinitely many twin primes? J. G. van der Corput (1939): There are infinitely many triples of primes in arithmetic progression. Ben Green and Terence Tao (2004): There exist sequences of primes in arithmetic progression of any given length.
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6 RSA Cryptography (after Rivest, Shamir and Adleman, 1977; slightly earlier: Ellis, Cocks and Williamson of the British Secret Service) - Choose two large prime numbers (of 100 digits, say), that’s the secret key. - Form their product (a 200-digit number), that’s the public key. - Use the public key to encrypt messages. - Decoding is possible only with the secret key.
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7 CONCLUSIONS 1 - Mathematics has beauty. - Ancient notions and proofs are as fresh today as 23 centuries ago. -
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This note was uploaded on 11/11/2011 for the course MATH 112 taught by Professor Jarvis during the Winter '08 term at BYU.

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PublicLecture - Mathematics: Beauty and the Beast Walter...

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