Download Finite Theorem of Arithmetic

By: Katrina Carlsen and Melissa Sparow

For every integer n ≥ 2, n is a prime number or
can be written uniquely (ignoring ordering) as a
product of prime numbers

Ex: unique factorization of 825

Divide by successively larger primes (2,3,5 etc)
Prove the factorization of a composite number
n>2 into prime numbers is unique when order is
not important.
N=p1p 2 p3 …pr
and
N=q1q 2 q3 …qs
p1p 2 p3 …pr = q1q 2 q3 …qs
P1| p1p 2…pr
so
This means
P1| q1q 2…qs
P1| qi
Contradiction. This means factorization is
unique!

integers a and b are relatively prime if
gcd(a,b) = 1.
 ia

+jb = 1
i(21) + j(16) = 1
Find i and j

i (27) + j(25) = 1

p|p and p|a


gcd(a,p)= p
gcd(a,p) = 1
 p|ab



P must divide either a or b
P doesn’t divide a..
1 = ia + jp

Linear combination of prime #s
Given positive integers a and b, gcd(a,b) is
the linear combination of a and b that has
the smallest positive value.
1
= ia + jp
 Multiply

b = (ia)b + (jp)b = i(ab) + (jp)b
 p|ab…


by b
ab written as kp
b = i(kp) + (jp)b
b = (ik +jb)p
 p|b
PROVEN!
 Our
goal was to proof the Fundamental
Theorem of Arithmetic
 In order to do so, we needed to understand
the Theorem on Division by Prime Numbers,
what it means to be Relatively Prime, Linear
Combinations and the Theorem on gcd(a,b).
Theorem on Size of Prime Factors: If n is a
composite number, then it has a prime factor
less than or equal to √n.
Given n=1021, find the prime factors of n or
determine that it is prime.
√1021 = 31.953
So we will test with prime numbers:
2,3,5,7,11, 13,17,19,23,29,31
None divides 1021. That means 1021 is prime.