Fundamental Theorem of Arithmetic
Even though this is one of the most important results in all of
Number Theory, it is rarely included in most high school
syllabi (in the US) formally. Interestingly enough, almost
everyone has an intuitive notion of this result and it is almost
always informally covered in middle school mathematics
classes in the United States.
The Fundamental Theorem of Arithmetic simply states that
each positive integer has an unique prime factorization. What
this means is that it is impossible to come up with two distinct
multisets of prime integers that both multiply to a given
positive integer.
To prove this, we must show two things:
1) Each positive integer can be prime factorized.
2) Each prime factorization is unique.
To see the first fact, let m>1 be the smallest positive integer
which does NOT have a prime factorization. Since m is not a
prime number, we can write m as a product of two factors, m
1
and
m
2
. But, since both of these are smaller than m, they DO
have prime factorizations. Thus, m can be expressed as the
product of these two factorizations, which creates a prime
factorization contradicting the assumption that m does not
have one.
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View Full DocumentBefore I continue with the second part of this proof, I want to
introduce pi notation, which is very similar to sigma notation:
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1
(
*
...
*
)
3
(
*
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2
(
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1
(
)
(
1
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f
n
f
f
f
f
i
f
n
i

=
∏
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The only difference between pi notation and sigma notation is
that each designated term is multiplied instead of added.
Also, I want to prove the two following lemmas:
1) If p is prime and a and b are positive integers and p  ab,
then either p  a or p  b.
2) If p is prime and a
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 Spring '09
 Number Theory, Integers, Prime number, prime Factorization, Fundamental theorem of arithmetic

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