primes - umbccampbell/mepp/primes/primes.pdf · the number of primes less than x is approximately...
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11/29/2010 1
Primes Rational, Gaussian, Industrial Strength, etc
Robert Campbell
11/29/2010 2
Primes and …
Theory – Number Theory to Abstract Algebra
History – Euclid to Wiles
Computation – pencil to supercomputer
Practical Uses – Cryptography, Error Correcting
Codes, etc
11/29/2010 3
Everybody Knows …
Everybody knows what a prime is: 2, 3, 5, 7, 9, 11, …
p is prime if its only positive divisors are 1 and p
p is prime if, whenever p divides ab, then either p
divides a and/or p divides b
Any number N factors into a product of primes
uniquely (up to order)
p is prime if, whenever p divides ab, then either p
divides a and/or p divides b
Any number N factors into a product of primes
uniquely (up to order)
11/29/2010 4
Primal Questions
Definition
Counting
Finding and Identifying
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Definition:
p is prime if its only positive divisors are 1 and p
p is prime if, whenever p divides ab, then either p divides a
and/or p divides b
Definition(s)
Definition:
p is irreducible if its only positive divisors are 1 and p
p is prime if, whenever p divides ab, then either p divides a
and/or p divides b
Thm: If p is prime then p is irreducibleLet a be a divisor of p, so p=ab for some b
Then p divides a and/or p divides b (as p is prime)
Case 1: p divides a. So a=pc, hence a=abc, so 1=bc and b=1. Thus a = p.
Case 2: p divides b. Similar argument - thus b = p and a = 1
Thm: If p is irreducible then p is prime
Proof requires division algorithm Euclidean Algorithm
11/29/2010 6
Division & Euclidean Algorithms
Division Algorithm/Property
Given positive integers a and b there are integers r and q with
a = bq + r and 0 ≤ r < b
Euclidean Algorithm
Given a and b, compute their greatest common divisor
(efficiently)
11/29/2010 7
Euclidean Algorithm (II)
Example - Compute gcd of 120 and 222:120 222
120 222-120=102
120-102=18 102
18 102-5*18=12
18-12=6 12
6 12-2*6=0
11/29/2010 8
Finding Primes
Sieve of Eratosthenes
1 2 3 4 5 6 7 8 9 10 11 12
13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30
31 32 33 34 35 36
1 2
Primes: 2
1 2 3 4 5 6 7 8 9 10 11 12
13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30
31 32 33 34 35 36
1 2 3 4 5 6 7 8 9 10 11 12
13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30
31 32 33 34 35 36
Primes: 2, 3
1 2 3 4 5 6 7 8 9 10 11 12
13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30
31 32 33 34 35 36
Primes: 2, 3, 5
1 2 3 4 5 6 7 8 9 10 11 12
13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30
31 32 33 34 35 36
Primes: 2, 3, 5, 7
1 2 3 4 5 6 7 8 9 10 11 12
13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30
31 32 33 34 35 36
Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31
11/29/2010 9
Counting Primes
There are an infinite number of prime numbers.
Proof: Assume not. So p1, … , pn is a list of all primes.
Then construct N = p1•…• pn +1 and note that none of the known
primes divides it. Thus N is prime – we have a contradiction.
Thus our assumption is incorrect – there are infinite primes.
Can we do better?
11/29/2010 10
Counting Primes (II)
π(x) := The number of primes no greater than x
π(10) = #{2, 3, 5, 7} = 4
π(20) = #{2, 3, 5, 7, 11, 13, 17, 19} = 8
π(100) = 25; π(1000) = 168; π(10000) = 1229; π(100000) = 9592;
Prime Number Theorem (Conjecture)
The number of primes less than x is approximately x/log(x)
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Counting Primes (III)
Conjectured
Gauss (1791? First published 1863)
Legendre (1798)
Proven
Hadamard (1896)
de la Vallée Poussin (1896)
Further work – Riemann Hypothesis
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Identifying Primes
Proofs – Given a number prove that it is prime
Tests – “Industrial Strength Primes”
11/29/2010 13
Identifying Primes (II)
[Fermat’s Little Theorem] If p is prime and p does not divide a,
then p divides (ap – a)
Proof: [Simple, but not today]
35 is not prime as:
Let p = 35 and a = 2
Compute (235 – 2) = 34359738366
Note that 35 does not divide 34359738366
17 might be prime as:
Let p = 17 and a = 3
Compute (317 – 3) = 129140160
Note that 17 divides 129140160 (in fact 129140160 = (17)(7596480))
11/29/2010 14
Efficiency Questions
Even/Odd Arithmetic:
Modular Arithmetic & Russian Peasants
+ Even Odd
Even Even Odd
Odd Odd Even
* Even Odd
Even Even Even
Odd Even Odd
Modular (Residue) Arithmetic: [Example: Mod 5]
+ 0 1 2 3 4
0 0 1 2 3 4
1 1 2 3 4 0
2 2 3 4 0 1
3 3 4 0 1 2
4 4 0 1 2 3
* 0 1 2 3 4
0 0 0 0 0 0
1 0 1 2 3 4
2 0 2 4 1 3
3 0 3 1 4 2
4 0 4 3 2 1
11/29/2010 15
Identifying Primes (III)
101 might be prime as:Let p = 101 and a = 2
Compute 2101 (mod 101)21 = 2
22 = (21)2 = 4
24 = (22)2 = 42 = 16
28 = (24)2 = 162 = 256 = 54 (mod 101)
216 = (28)2 = 542 = 2916 = 88 (mod 101)
232 = (216)2 = 882 = 7744 = 68 (mod 101)
264 = (232)2 = 682 = 4624 = 79 (mod 101)
2101 = 264+32+4+1 = (264)(232)(24)(21)=(79)(68)(16)(2)=171904 = 2 (mod 101)
Try this for 5446367
But … try this for 561
11/29/2010 16
Open Questions
Find integers a, b, c and
n>2 with an + bn = cn (FLT)
Any even integer greater than 2 is the sum of
two primes (Goldbach) [e.g. 36 = 29 + 7]
Are there an infinite number of successive
odd numbers which are prime? (Twin Prime)
[e.g. {3,5}, {5,7},…, {281, 283}, …]
Is there a prime of the form p = 22n + 1 for
n>4? (Fermat Prime) [e.g. F3 = 223 + 1 = 257]
11/29/2010 17
Extension: Gaussian Integers
Consider the complex numbers with integer
coefficients: {n + mi} = Z[-1]
All the “nice properties” hold:
There are an infinite number of irreducibles: 3, 1 i, 7, 2 i, …
Unique factorization into irreducibles (up to order and
multiples of i and 1)
We can sieve to find primes
We can test for primality
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A GI Eratosieve
0-1 1
-i
i
1 i
2 I, 1 2i
3, 3i
3 2i, 2 3i,
…
11/29/2010 19
Identifying GI Primes
Given a prime p, can we test to see if it is prime?
Fermat’s Little Thm (extended to Gaussian Integers)
If p is a Gaussian prime, and p does not divide a, then p
divides (aN(p) – a), where N(p) = pp* is the norm of p.
Examples:
(1+i)49 – (1+i) = 0 (mod 7), so 7 is probably a Gaussian prime
(2+i)13 – (2+i) = 0 (mod 2+3i), so 2+3i is probably a GI prime
(2+i)34 - (2+i) = 1 + 3i (mod 5+3i), so 5+3i is not a GI prime
11/29/2010 20
A Counterexample
Consider Z[-6] = {n + m -6}
Find primes by sieving
The only units are 1
2, 3, -6, 1+ -6, … are irreducible
But …
6 = (-1)(-6)2
6 = (2)(3)
Factorization is not unique
-6 divides 6 and 6 = (2)(3), but -6 divides neither 2 nor 3
Z[-6] has irreducibles, but no primes
11/29/2010 21
References & Further Reading
The Elements, Euclid (ca 300 BC) (trans Thomas Heath), Dover Publ
http://aleph0.clarku.edu/~djoyce/java/elements/elements.html
Elementary Number Theory, Jones & Jones, Springer, 1998
The Book of Numbers, Conway & Guy, Copernicus, 1996
Prime Numbers, A Computational Perspective, Crandall & Pomerance, Telos Publ, 2001
Factorization and Primality Testing, Bressoud, Springer, 1989
Algebraic Number Theory and Fermat’s Last Thm, 3rd Ed, Stewart & Tall, Peters Publ, 2002
The Primes Pages http://www.utm.edu/research/primes/
Computational Number Theoryhttp://www.math.umbc.edu/~campbell/NumbThy/Class/