PublicLecture

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

This preview shows pages 1–8. Sign up to view the full content.

1 Mathematics: Beauty and the Beast Walter Tholen York University Toronto

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.)
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.”

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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!
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
7 CONCLUSIONS 1 - Mathematics has beauty. - Ancient notions and proofs are as fresh today as 23 centuries ago. -

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 20

PublicLecture - Mathematics Beauty and the Beast Walter...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online