Expert: Paul Klarreich - 10/5/2010

Question
I need some help w/this proof: "If a whole number is of the form 6n-1, then it must have a prime factor other than 2 or 3."  So far I have considered the case where 6n-1 is prime (then clearly it has a prime factor greater than or equal to 5--that is, 6n-1 itself); however, I don't know what to do when 6n-1 is composite (I think it might involve writing 6n-1 = (6a+1)(6b-1) = 6(6ab-a+b)-1 and showing that 6n-1 is a product of twin primes.  I need help finishing this one off though.  Thanks a lot!

Answer
Questioner:   Dean
Country:  United States
Category:  Advanced Math
Private:  No
 
Subject:  Number theory
Question:  I need some help w/this proof: "If a whole number is of the form 6n-1, then it must have a prime factor other than 2 or 3."  So far I have considered the case where 6n-1 is prime (then clearly it has a prime factor greater than or equal to 5--that is, 6n-1 itself); however, I don't know what to do when 6n-1 is composite (I think it might involve writing 6n-1 = (6a+1)(6b-1) = 6(6ab-a+b)-1 and showing that 6n-1 is a product of twin primes.  I need help finishing this one off though.  Thanks a lot!

...........................
Now  6n - 1 is NOT divisible by 2
and  6n - 1 is NOT divisible by 3

But by the FTA, it must have a prime factorization.  Since the prime factors cannot include 2,3,  then....

Can you finish?