induction proof of fundamental theorem of arithmetic

induction proof of fundamental theorem of arithmetic

We present an

induction

proof by Zermelo for the

fundamental theorem of arithmetic

(

http://planetmath.org/FundamentalTheoremOfArithmetic

).

Part 1.

Every

positive integer

n

is a

product

of

prime numbers

.

Proof.

If

n

=

1

, it is the

empty product

of primes, and
if

n

=

2

, it is a prime number.

Let then

n

>

2

.  Make the

induction hypothesis

that all positive
integers

m

with

1

<

m

<

n

are products of prime numbers.
If

n

is a prime number, the thing is ready.  Else,

n

is a
product of smaller numbers; these are, by the induction hypothesis,
products of prime numbers.  The proof is

complete

.

Part 2.

For any positive integer

n

, its

representation

as product of prime numbers is unique up to the order of the

prime factors

.

Proof.

The assertion is clear in the case that

n

is a
prime number, especially when

n

=

2

.

Let then

n

>

2

and suppose that the assertion is true for
all positive integers less than

n

.

If now

n

is a prime, we are ready.  Therefore let it be a

composite number

.  There is a least nontrivial factor

p

of

n

.
This

p

must be a prime.  Put

n

=

p

⁢

b

where  b is a
positive integer.  By the induction hypothesis,

b

has a

unique prime

factor decomposition.  Thus

n

has a unique prime
decomposition containing the prime factor

p

.

Now we will show that

n

cannot have other prime decompositions.
Make the antithesis that

n

has a different prime decomposition;
let

q

be the least prime factor in it.  Now we have

p

<

q

and

n

=

q

⁢

c

where

c

∈

ℤ

+

and

c

<

n

with

p

∤

c

.  Then

n

0

:=

n

-

p

⁢

c

=

{

p

⁢

b

-

p

⁢

c

=

p

⁢

(

b

-

c

)

q

⁢

c

-

p

⁢

c

=

(

q

-

p

)

⁢

c

is a positive integer less than

n

.  Since

p

∣

n

0

, the
induction hypothesis implies that the prime

p

is in the prime
decomposition of

(

q

-

p

)

⁢

c

and thus also at least of

q

-

p

or

c

.  But we know that

p

∤

c

, whence

p

∣

q

-

p

.
Thus we would get

p

∣

q

-

p

+

p

=

q

.  Because both

p

and

q

are primes, it would follow that

p

=

q

.  This
contradicts the fact that

p

<

q

.  Consequently, our
antithesis is wrong.  Accordingly,

n

has only one prime
decomposition, and the induction proof is complete.

References

1

Esa V. Vesalainen

: “Zermelo ja aritmetiikan
peruslause”.

-

Solmu

1

(2014).

2

Ernst Zermelo

:

Elementare
Betrachtungen zur Theorie der Primzahlen

.

-

Wissenschaftliche Gesellschaft zu
Göttingen (1934). English translation in:

3

H.-D. Ebbinghaus & A. Kanamori (eds.):

Ernst Zermelo. Collected
Works. Volume I.

Set Theory

, Miscellanea, Springer (2010). Ernst Zermelo: “

Elementary

considerations concerning the theory of prime numbers” 576

-

581.

Title

induction proof of fundamental theorem of arithmetic

Canonical

name

InductionProofOfFundamentalTheoremOfArithmetic

Date of creation

2015-04-08 7:32:53

Last modified on

2015-04-08 7:32:53

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Proof

Generated on Fri Feb 9 09:58:31 2018 by

LaTeXML