Chapter 3: Primes and their Distribution
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Chapter 3: Primes and their Distribution
SECTION A Introduction to Primes
By the end of this section you will be able to
• understand the importance of primes
• prove properties of primes
A1 Importance of Primes
Prime numbers are central to number theory. We first define what is meant by a prime
number and then state and prove some properties of prime numbers.
Definition (3.1).
An integer p greater than 1 is called a prime number (prime) if its only divisors are 1 and
p. An integer greater than 1 that is not prime is called composite.
This definition means that every integer greater than 1 is either prime or composite.
Examples of prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, …
Examples of composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, …
The only even prime is 2. Why are primes important?
Because the ‘Fundamental Theorem of Arithmetic’ states that:
Every integer greater than 1 is either a prime or a product of primes whose representation is
unique apart from the order.
Examples of this are
3 = 3, 10 = 2 × 5, 20 = 22 × 5, 100 = 22 × 52 , 101 = 101, …
The Fundamental Theorem of Arithmetic says that the product of primes is unique apart
from the order. What does this mean?
If we consider 100 = 22 × 52 then the 2 and 5 are the only primes which multiplied together
a number of times gives 100. There will be no other primes in the decomposition of 100
into primes. Of course we can write 100 = 22 × 52 = 2 × 2 × 52 = 5 × 5 × 2 × 2 = ...
However this is just changing the order of multiplication. The prime numbers 2 and 5 are
the building blocks of 100.
100
22
2
2
Fig 1
This figure shows the building block for 100.
52
5
5
Chapter 3: Primes and their Distribution
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This fundamental theorem says that every integer greater than 1 is either a prime number or
can be made up by a product of primes. This means that primes are a building block of the
positive integers.
A2 Properties of Primes
How do we know that primes are multiples of positive integers?
We examine the divisors of a positive integer greater than 1. For example
2 20 implies that 2 ( 4 × 5 ) which implies 2 4
Note that 2 is a prime number. In general we have:
Proposition (3.2).
If p is prime and p ab then p a or p b .
Proof.
Suppose p does not divide a. This means that gcd ( a, p ) = 1 . Using Euclid’s Lemma
which is (2.5) in Burton:
If a bc and gcd ( a, b ) = 1 then a c
Applying this to p ab with gcd ( a, p ) = 1 gives p b .
Similarly if p does not divide b then applying Euclid’s Lemma gives p a .
Either way we have our result.
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We can extend this proposition to more than multiplication of two terms ab :
Corollary (3.3).
If p is prime and p a1a2 a3 L an then p a1 or p a2 or … or p an .
Proof.
We can prove this result by using induction.
Step 1 : With p a1a2 gives p a1 or p a2 by the above Proposition (3.2).
Step 2 : Assume the result is true for n = k :
p a1a2 a3 L ak implies that p a1 or p a2 or … or p ak
Step 3 : Required to prove this result for n = k + 1 :
p a1a2 a3 L ak +1 implies that p a1 or p a2 or … or p ak +1
We have p a1a2 a3 L ak ak +1 which means that
p
( a1a2 a3 L ak ) ak +1
Applying the above Proposition (3.2) we have
p ( a1a2 a3 L ak ) or p ak +1
Using step 2 on p
( a1a2 a3 L ak )
gives p a1 or p a2 or … or p ak . Hence we have our
result:
p a1a2 a3 L ak +1 implies that p a1 or p a2 or … or p ak +1
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If the a’s in this Corollary are prime then p is equal to one of the a’s. For example
Chapter 3: Primes and their Distribution
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7 3 × 5 × 7 × 11× 13
This is always true as the next result states:
Corollary (3.4).
If p, q1 , q2 , q3 , L , qn are all primes and p
k, where 1 ≤ k ≤ n .
Proof.
We are given that p
( q1 × q2 × q3 × L × qn )
( q1 × q2 × q3 × L × qn )
then p = qk for some
where q’s are prime. By the above Corollary
(3.3) on page 2:
If p is prime and p a1a2 a3 L an then p a1 or p a2 or … or p an .
we can say that p qk . Since qk is prime the only divisors of this are 1 and qk . Hence this
means that p = qk because p is prime.
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A3 Fundamental Theorem of Arithmetic
This is a powerful result in mathematics.
Fundamental Theorem of Arithmetic (3.5).
Every positive integer n greater than 1 is either a prime or can be written uniquely as a
product of primes apart from the order.
How do we prove this result?
First we prove that n is a product of primes and then we show that this representation is
unique apart from the order.
Proof.
Proof that n is product of primes:
Either n is prime or composite.
If n is a prime then we are done.
If n is composite then it has a divisor, say d, which means that d n . Amongst all the
divisors d there must be a smallest divisor, call this p1 , of n. Why?
Because of the Well Ordering Principle on page 1 of Burton:
Every non-empty subset of non-negative integers has a least element.
This p1 must be prime otherwise we would have a smaller divisor of n. Since p1 n we
can write n as
n = p1n1 where n1 is an integer
If n1 is prime then we have shown that n is a product of primes and only need to prove
uniqueness.
If n1 is composite then we can repeat the above process:
Let p2 be the smallest divisor of n1 and as above p2 must be prime. Hence p2 n1 so
n1 = p2 n2 where n2 is an integer
Substituting this n1 = p2 n2 into the above n = p1n1 gives
n = p1 p2 n2
If n2 is prime then we have our product of primes. If n2 is composite then repeating the
above process we have
n = p1 p2 p3 n3
Chapter 3: Primes and their Distribution
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This cannot continue forever there must be an integer nk say, where nk is prime, that is
nk = pk . We have
n = p1 p2 p3 L nk = p1 p2 p3 L pk
Hence we have shown that n this is a product of primes.
Uniqueness:
Suppose that
n = p1 p2 p3 L pr = q1q2 q3 L qs
where the p’s and q’s are prime and they are in ascending order, that is
p1 ≥ p2 ≥ p3 ≥ L ≥ pr and q1 ≥ q2 ≥ q3 ≥ L ≥ qs
Without loss of generality assume s ≥ r .
This p1 p2 p3 L pr = q1q2 q3 L qs means that p1 ( q1q2 q3 L qs ) . Using the above Corollary
(3.4):
If p, q1 , q2 , q3 , L , qn are all primes and p
( q1 × q2 × q3 × L × qn )
then p = qk for some
k, where 1 ≤ k ≤ n .
on this p1
( q1q2 q3 L qs )
yields
p1 = qk for some k, where 1 ≤ k ≤ s
This means that p1 ≤ q1 .
( p1 p2 p3 L pr ) and again by Corollary (3.4):
Similarly q1
q1 = pm for some m, where 1 ≤ m ≤ r
This means that q1 ≤ p1 . Combining the two results p1 ≤ q1 and q1 ≤ p1 gives
p1 = q1
Again repeating this process we obtain
p2 = q2 , p3 = q3 , p4 = q4 , …, pr = qr
If s > r then by cancelling out the common factors in p1 p2 p3 L pr = q1q2 q3 L qs gives
1 = qr +1qr + 2 qr +3 L qs
This is impossible because the q’s are primes. Hence s = r which means that
p1 = q1 , p2 = q2 , p3 = q3 , p4 = q4 , …, pr = qr
The factorization of n is unique.
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Note that the prime factors may repeat. For example
120 = 2 × 2 × 2 × 3 × 5 = 23 × 3 × 5
We can also write Fundamental Theorem of Arithmetic as:
Corollary (3.6).
Every positive integer n greater than 1 is either a prime or can be written uniquely as a
product of primes apart from the order in the following manner:
n = p1k1 p2 k2 p3k3 L pr kr
For example
360 = 2 × 2 × 2 × 3 × 3 × 5 = 23325
1 000 000 = 2656
1 000 001 = 101× 9901
2789865215 = 5 × (557973043)