€
€
€
2. Prime numbers
1
Here is another characterization of prime numbers.
Theorem p is prime ⇔ it has no divisors d that satisfy
1< d ≤ p .
Proof [ ⇒] If
€ p is prime then it has no divisors d that
satisfy 1 < d < p, so clearly no divisor of p satisfies
1< d ≤ p .
€ [ ⇐] If the number p has no divisors d that satisfy
1 < d ≤ p , then its only possible divisors, other than 1,
€
are greater than p . So if p were composite, we could
€ find divisors a and b for which p = ab > p ⋅ p = p,
which is absurd. Therefore, p must be prime. //
€
€ technique for determining
This provides us with a
whether a number p is prime: perform trial division of p
by every number from 2 up to p . If a divisor is not
found, then p must be prime. Indeed, only prime
numbers need to be tested as potential factors, since
any composite divisor will be reached only after all its
€ the same test.
prime factors have passed
This is the basis for construction of the Sieve of
Eratosthenes, an algorithm for listing the primes.
From a list of the integers from 2 to n, we note that the
first number must be prime. Then strike from the list
€
2. Prime numbers
2
all its multiples (4, 6, 8, …). The smallest number not
struck (3) must then be prime. Next, strike from the
list all the multiples of 3 that have not already been
struck (9, 15, 21, …). The smallest number not struck
(5) must then be prime. Continue in this fashion until
you have found the primes up to n . All the remaining
numbers between n and n which have not been struck
must all be primes. (Why?)
€
Will we ever run out of primes? Well, ... no.
€
Theorem There are infinitely many primes.
Proof [Euclid] Suppose there were only finitely many
primes. List them as p1, p2 ,…, pn . Then the number
N = p1 p2  pn +1, which is clearly larger than all the
primes, must be composite. So it must have a prime
divisor, but none of the primes can divide it, since by
€
the DA, division of N by any of the p’s leaves remainder
1. So there must be infinitely many primes. //
Another Proof! [Euler] Since 1p < 1 for every prime p,
we can use the geometric series formula to write
 1 k
1
∑€
.
  =
1

p

1−
k≥ 0
p
€
2. Prime numbers
3
For example,
1
= 1+ 12 + 14 + 18 +
1
1− 2
1
1 + 1 + 1 +
=
1+
3 9 27
1− 13
1
1 + 1 + 1 +
=
1+
5 25 125
1− 15
and so on. Multiplying these (convergent) series
together, we get
€
1
1
1 + 1 + 1 + 1 + 1 + 1 + 1 +
⋅
=
1+
2 3 4 6 8 9 12
1 − 12 1− 13
1
1
1
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 +
⋅
⋅
=
1+
2 3 4 5 6 8 9 10 12
1− 12 1 − 13 1− 15
and so on. So
€
1
1
1
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 +
⋅
⋅
⋅
=
1+
2 3 4 5 6 7 8 9 10 11 12
1− 12 1 − 13 1− 15
which is more compactly written as
€
∏
all primes p
€
1
1 − 1p
=
1
n ≥1 n
∑
2. Prime numbers
4
The series on the right side is the (famous) harmonic
series, which is well-known to be a divergent series. It
follows that the product on the left cannot have only
finitely many factors. //
Number theorists have long studied the prime number
function
π (x) = #(primes p ≤ x)
(e.g., π (100) = 25) and the nth prime function
€
p(n) = (the nth prime number)
€ (e.g., p(100) = 499) with the hope of discovering a
pattern to the growth of the prime numbers. (The
€
functions
are related directly by the fact that
π ( p(n)) = n .) These patterns are elusive and subtle. For
instance, it is known that p(n) is not representable as a
polynomial function, nor is π (x).
€
€
2. Prime numbers
5
Here is some illustrative data on π (x):
x
€
π (x) / x
x / π (x)
ln x
0.400
2.500
2.303
0360
2.778
3.219
15
0.300
3.333
€
3.912
25
€
0.250
4.000
4.605
200
46
0.230
4.348
5.298
500
95
0.190
5.263
6.215
1000
168
0.168
5.952
6.908
10000
1 229
0.134
8.137
9.210
1000000
78 498
0.078
12.739
13.816
1000000000
50 847 534
0.050
19.667
20.723
π (x)
€
10
4
25
9
50
100
€
€
Note first that the percentage of primes within the
range of numbers from 1 to x decreases steadily as x
increases; that is, primes become rarer and rarer as
they get larger.
The last two columns indicate that the quantity x / π (x)
seems to grow logarithmically. In fact, Gauss
conjectured precisely this in the 1790s. It took 100
years of concerted effort to prove it, and the final
€ from complex
discovery required sophisticated methods
analysis to be realized!
2. Prime numbers
6
The Prime Number Theorem [Hadamard & de la
Vallee Poussin, 1896]
π (x)
= 1 . //
x → ∞ x ln x
lim
Before we proceed, let’s consider a number of curious
facts about €
the set of primes. A pair of prime numbers
whose difference is 2 is called a twin prime pair (e.g.,
3 and 5, 5 and 7, 41 and 43, 1997 and 1999, 2027 and
2029, … ). A famous unsolved problem in mathematics
is the Twin Prime Conjecture, which states that there
are infinitely many twin prime pairs; there is no known
proof of this, despite strong evidence that it is true. For
instance, in 1919 Viggo Brun showed that the series
∑
twin primes p
1
p
must converge. (Of course, it still might be an infinite
series!) Also, it is conjectured that if
€
T(x) = #(primes p ≤ x so that p + 2 is also prime),
then
lim
x →∞
€
T(x)
2 = 0.66016...
x (ln x)
2. Prime numbers
7
Another famous outstanding unsolved problem is the
The Goldbach Conjecture Every even number
greater than 2 is the sum of two primes. //
It has been exceedingly difficult to make headway
towards proving this result. But we will cite two
partial results, both of which are quite hard to prove:
Theorem [Vinogradov, 1937] Every sufficiently large
odd number is the sum of three primes. (That is, there
exists some integer N, whose specific value is not
known, so that every odd number greater than N is the
sum of three primes.) //
Theorem [Chen, 1966] Every sufficiently large even
number is the sum of two numbers one of which is
prime and the other of which is the product of at most
two primes. (Again, this means that there exists some
integer N, whose specific value is not known, so that
every even number greater than N is the sum of two
numbers one of which is prime and the other of which is
the product of at most two primes.) //