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Mathematics:
Beauty and the Beast
Walter Tholen
York University
Toronto
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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.”
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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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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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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 200digit 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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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.
 Winter '08
 JARVIS
 Prime Numbers

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