fundamental theorem of arithmetic, proof of the

fundamental theorem of arithmetic, proof of the

To prove the

fundamental theorem of arithmetic

, we must show that each

positive integer

has a prime decomposition and that each such
decomposition is unique up to the order (

http://planetmath.org/OrderingRelation

) of the factors. Before
proceeding with the proof, we note that in any

integral domain

, every
prime is an

irreducible element

. Furthermore, by Euclid’s lemma, every irreducible element in

ℤ

is prime. As a result,

prime elements

and irreducible elements are

equivalent

in

ℤ

. We will use this fact to prove the

theorem

.

Note: all our

arithmetic

will be carried out in the

natural numbers

, not the

integers

.

Existence

Now we prove the existence of a prime decomposition. Since 1 has a
prime decomposition and any prime has a prime decomposition, it
suffices to show that any

composite number

has a prime decomposition.
We claim that any composite number is

divisible

by some

prime number

.
To see this, assume

n

is a composite positive integer. Then
there exist positive integers

a

and

b

, necessarily strictly
smaller than

n

, such that

n

=

ab

. If

a

is not
prime we can write it as a

product

of smaller positive integers in the
same way. Continuing this process, we obtain a strictly decreasing

sequence

of natural numbers. By the

well-ordering principle

, this
sequence must terminate, and by construction, it must terminate in a
prime number. Hence

n

is divisible by a prime number, as
desired.

To

complete

the proof of existence, we apply the well-ordering
principle again. If

n

is composite, then it is divisible by some
prime. Moreover, the quotient is strictly smaller than

n

. If
the quotient is not prime then as before we find a prime that divides
it; continuing this process, we produce a strictly decreasing sequence
of natural numbers. By the well-ordering principle, this sequence
terminates in a prime number. The product of the prime numbers we
found in the construction of this sequence is simply

n

. Thus
every composite number has a prime decomposition.

Uniqueness

Now we show uniqueness up to order of factors. Suppose we are given
two prime decompositions

n

=

p

1

⁢

⋯

⁢

p

k

=

q

1

⁢

⋯

⁢

q

ℓ

of

n

, where

k

≤

ℓ

and the factors in each prime decomposition
are arranged in nondecreasing order. Since

p

1

divides

n

, it
divides the product

q

1

⁢

⋯

⁢

q

ℓ

, so by

primality

must divide
some

q

i

. Thus there is a natural number

b

such that

q

i

=

b

⋅

p

1

. Since

q

i

is an irreducible element and

p

1

is not a
unit, it must be that

b

is a unit, that is,

b

= 1. Hence

p

1

=

q

i

≥

q

1

. Similarly, there is some

j

such that

q

1

=

p

j

≥

p

1

. Since

p

1

≥

q

1

and

q

1

≥

p

1

, it follows that

p

1

=

q

1

. Cancelling these factors yields the simpler equation

p

2

⁢

⋯

⁢

p

k

=

q

2

⁢

⋯

⁢

q

ℓ

.

By repeating the above procedure we can show that

p

i

=

q

i

for all

i

from 0 to

k

. Cancelling these factors gives the equation

1

=

q

k

+

1

⁢

⋯

⁢

q

ℓ

.

Since a nontrivial product of primes is greater than 1, the right-hand
side of this equation is the empty product. We conclude that

k

=

l

. Hence all prime decompositions of

n

use the same number
of factors, and the factors which appear are unique up to the order in
which they appear. This completes the proof of uniqueness and thereby
the proof of the theorem.

Remark

. Note that Euclid’s Lemma is necessary in order to prove the uniqueness portion of the theorem. Also, an alternative way of proving the existence portion of the theorem is to use

induction

: if

n

is composite, then

n

=

a

⁢

b

for some integers

a

,

b

<

n

(this is true, for if one of

a

,

b

is

n

, then the other integer must be

1

). By induction, both

a

and

b

can be written as product of primes, which implies that

n

is a product of primes.

Title

fundamental theorem of arithmetic, proof of the

Canonical name

FundamentalTheoremOfArithmeticProofOfThe

Date of creation

2013-03-22 11:48:00

Last modified on

2013-03-22 11:48:00

Owner

mps (409)

Last modified by

mps (409)

Numerical id

16

Author

mps (409)

Entry type

Proof

Classification

msc 11-00

Classification

msc 16U99

Classification

msc 13A99

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