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COMPSCI 230 — Discrete Math
January 29, 2015
COMPSCI 230 — Discrete Math
Prime Numbers
January 29, 2015
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Outline
1
Prime Numbers
The Sieve of Eratosthenes
Racket Implementation with Lists
More on Primes
COMPSCI 230 — Discrete Math
Prime Numbers
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Prime Numbers
Primes
• Natural numbers greater than 1 with no
proper divisors
• 1, 2, 3, 6 | 6
but
1, 7 | 7
• Numbers in red are proper divisors
• Fundamental theorem of arithmetic:
1 < 𝑛 ∈ 𝐍 ⇒ 𝑛 = ∏ 𝑝𝑖
𝑖
where the 𝑝𝑖 s are prime
• Example: 26040 = 23 × 3 × 5 × 7 × 31
• Fundamental in number theory
• Applied in cryptography, electrical
communications, computer chips,...
COMPSCI 230 — Discrete Math
Prime Numbers
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Prime Numbers
Problems About Primes
• Generate all primes up to 𝑛 (sieve)
• Test whether 𝑛 is prime
• Is 802117 prime?
• Factor 𝑛 into primes
• 802117 = 821 × 977
• You did factor, which subsumes test, so
you know both
• There are much more efficient methods
COMPSCI 230 — Discrete Math
Prime Numbers
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Prime Numbers
The Sieve of Eratosthenes
The Sieve of Eratosthenes
To generate all primes up to 𝑛:
• List all integers from 2 to 𝑛
• Let 𝑝 = 2 (smallest prime)
• Cross 2𝑝, 3𝑝, … from the list
• First remaining number must be prime
• Set 𝑝 to that, and repeat until no more
numbers are left
COMPSCI 230 — Discrete Math
Prime Numbers
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Prime Numbers
The Sieve of Eratosthenes
Example: 𝑛 = 36
2
9
16
23
30
COMPSCI 230 — Discrete Math
3 4 5 6
10 11 12 13
17 18 19 20
24 25 26 27
31 32 33 34
Prime Numbers
7
14
21
28
35
8
15
22
29
36
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Prime Numbers
The Sieve of Eratosthenes
The Sieve of Eratosthenes
• Speedups exist
• Takes large amounts of storage for
nontrivial 𝑛
• Each number is touched at least once
• Takes at least 𝑛 steps
COMPSCI 230 — Discrete Math
Prime Numbers
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Prime Numbers
Racket Implementation with Lists
Racket Implementation:
Preliminaries
(build-list 3 (lambda (x) x))
(cons 1 ’(2 3 4))
(filter odd? ’(1 2 3 4))
(reverse ’(1 2 3))
→
→
→
→
(build-list 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛)
(cons 𝑖𝑡𝑒𝑚 𝑙𝑖𝑠𝑡)
(filter 𝑝𝑟𝑒𝑑𝑖𝑐𝑎𝑡𝑒 𝑙𝑖𝑠𝑡)
(reverse 𝑙𝑖𝑠𝑡)
COMPSCI 230 — Discrete Math
Prime Numbers
’(0
’(1
’(1
’(3
→
→
→
→
1 2)
2 3 4)
3)
2 1)
𝑙𝑖𝑠𝑡
𝑙𝑖𝑠𝑡
𝑙𝑖𝑠𝑡
𝑙𝑖𝑠𝑡
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Prime Numbers
Racket Implementation with Lists
Implementation Strategy
• build-list the integers → numbers
• Initially, the list of primes is empty
• (process primes numbers) will recursively
• (cons (first numbers) primes)
• (filter
<non-multiples of (first numbers)>
numbers)
• call itself with primes and the filtered numbers
• until (empty? numbers)
• and then return primes
• reverse the result if desired
COMPSCI 230 — Discrete Math
Prime Numbers
January 29, 2015
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Prime Numbers
Racket Implementation with Lists
Racket Code
(define (sieve n)
(define (process primes numbers)
(cond ((empty? numbers) primes)
(else
(process
(cons (first numbers) primes)
(filter
(lambda (x)
(not (= 0 (remainder x (first numbers)))))
numbers)))))
(reverse
(process ’() (build-list (- n 1) (lambda (x) (+ 2 x))))))
COMPSCI 230 — Discrete Math
Prime Numbers
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Prime Numbers
Racket Implementation with Lists
Output Sample
> (sieve 40)
’(2 3 5 7 11 13 17 19 23 29 31 37)
COMPSCI 230 — Discrete Math
Prime Numbers
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Prime Numbers
More on Primes
More on Primes
• We will use cryptography as an example
application of prime numbers
• RSA (Ron Rivest, Adi Shamir, Leonard
Adleman, 1978) cipher
• Used in sending credit card info, encrypted
email, file encryption, store passwords,
secure shell or ftp, much more
• We need a few preliminaries about primes
COMPSCI 230 — Discrete Math
Prime Numbers
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Prime Numbers
More on Primes
Co-Primes
• 𝑚 and 𝑛 are co-primes if they share no
•
•
•
•
•
prime factors
Not co-primes: 𝑚 = 24 = 23 × 3 and
𝑛 = 15 = 3 × 5
Co-primes: 𝑚 = 24 = 23 × 3 and
𝑛 = 35 = 5 × 7
Two distinct prime numbers are always
co-prime: 𝑚 = 13 and 𝑛 = 2
A prime is co-prime with all the naturals it
does not divide into
Example: 2 is co-prime with all the odd
numbers
COMPSCI 230 — Discrete Math
Prime Numbers
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