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The lore of prime numbers
P.J. Forrester
Australian Professorial Fellow
Supported by the Australian Research Council and
The University of Melbourne
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General audience references:
“Dr Riemann’s zeros” by Karl Sabbagh (Atlantic Books, 2002).
“Prime Obsession: Bernhard Riemann and the greatest unsolved
problem in mathematics” by John Derbyshire (Joseph Henry Press,
2003).
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Primes
• Whole (natural) numbers greater than or equal to 2, which cannot
be factorized into a product of smaller whole numbers.
2 3 5 7 11 13 17 19 23 29
...
• (Euclid) Every natural number can be factored uniquely into a
product of primes.
24 = 4 × 6 = 3 × 8 = 23 × 3.
• (Euclid) There are infinitely many primes.
Proof by contradiction.
Suppose the list of primes was finite.
2
3 5 7
Argue that then
2×3×5×7+1
is prime but not in the list.
This is a contradiction.
Hence the list of primes in not finite.
Remark
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2 × 3 × 5 × 7 × 11 × 13 + 1 = 30031 = 59 × 509.
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Listing primes: the sieve of Eratosthenes
Write down the numbers 1 to 100.
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 37
38 39
40
41 42 43 44 45 46 47
48 49
50
51 52 53 54 55 56 57
58 59
60
61 62 63 64 65 66 67
68 69
70
71 72 73 74 75 76 77
78 79
80
81 82 83 84 85 86 87
88 89
90
91 92 93 94 95 96 97
98 99 100
Remove from this list all multiples of the first prime number 2.
11
21
31
41
51
61
71
81
91
2
3
·
13 · 15 · 17
· 19 ·
33 · 35 · 37
· 39 ·
·
·
·
·
·
·
·
·
·
5
·
7
23 · 25 · 27
43 · 45 · 47
53 · 55 · 57
63 · 65 · 67
73 · 75 · 77
83 · 85 · 87
93 · 95 · 97
·
9
·
· 29 ·
· 49 ·
· 59 ·
· 69 ·
· 79 ·
· 89 ·
· 99 ·
Now remove all remaining multiples of the second prime number 3.
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2
3
11
·
13 ·
·
·
31
·
41
·
·
·
61
·
71
·
·
·
91
·
·
7
23 · 25 ·
·
·
·
5
· 17
·
· 35 · 37
43 ·
· 47
·
53 · 55 ·
·
· 65 · 67
73 ·
· 77
·
83 · 85 ·
·
·
·
· 95 · 97
·
·
·
· 19 ·
· 29 ·
·
·
·
· 49 ·
· 59 ·
·
·
·
· 79 ·
· 89 ·
·
·
·
Continue in this fashion, removing all multiples of the prime numbers
up to ten: 2, 3, 5, 7.
2
3
11
·
13 ·
·
·
31
41
·
61
71
·
·
·
·
·
·
·
·
·
· 5 ·
23 ·
·
·
43 ·
53 ·
·
·
73 ·
83 ·
·
·
7
·
·
·
·
· 17 · 19 ·
·
·
·
· 29 ·
· 37 ·
·
·
· 47 ·
·
·
·
·
·
·
·
·
· 59 ·
·
· 67 ·
·
·
·
·
·
·
· 79 ·
· 89 ·
· 97 ·
·
·
·
·
This leaves us with all the primes up to 100. We count 25 such numbers.
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One (of many) unsolved problems about primes
• (Goldbach’s conjecture) Is every even integer n ≥ 4 the sum of two
primes?
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4=2+2
6=3+3
8=3+5
10 = 5 + 5
12 = 5 + 7
14 = 7 + 7
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“How frequent are the primes?”
• Out of the first 102 natural numbers, 25 are prime: 1 in 4.
• Out of the first 104 natural numbers, 1229 are prime: 1 in 8.1.
• Out of the first 107 natural numbers, 664,579 are prime: 1 in 15.
• Data suggest — up to 10n , about one in every 2.3n are prime.
Legendre’s logarithmic law
Out of the first N natural numbers, roughly 1 in log e N are prime.
Gauss’ improvement
• The density of primes about the number N is approximately
Hence, out of the first N natural numbers, approximately
Z N
dt
loge t
2
1
loge N .
are prime.
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Graph of
Z
|
N
2
dt
log t
{z e }
−(number of primes less than or equal to N )
logarithmic integral
as a function of N .
150
125
100
75
50
25
50
100
150
units of 10^4
200
The Riemann hypothesis
• The number of primes less than or equal to N , for N large, is given
by
logarithmic integral + correction term
where
correction term ∝
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√
N loge N.
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Interpretation and consequence
•
√
N corrections are familiar in probability theory.
• Kramer’s model — Statistical properties of primes are well
described by the statement that each positive integer n ≥ 2 is a
prime with probability 1/ log e n.
Numerical experiment
List say 2, 000 primes starting from the first prime bigger than 10 9 .
These are the numbers 109 + x with x equal to
7, 9, 21, 33, 87, 93, 97, 103, · · ·
What is the distribution of the gap between primes?
p^{(N)}(0;s)
p^{(N)}(1;s)
0.7
1
0.6
0.8
0.5
0.6
0.4
0.3
0.4
0.2
0.2
0.1
1
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4
8
s
1
2
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4
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s
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