homework-6

Question

Answer 1

Proof by contradiction:
Assume n,n+2, n+4 are all prime.

Using the hint given in the question that every integer can be expressed as 3x OR 3x+1 OR 3x+2 where x is an integer
we can gave 3 cases

CASE 1:
n mod3 = 0
This makes n non-prime.
Hence the contradiction.
 

CASE 2:
n mod3 = 1
then (n+2) mod3 = 0
this makes number n+2 non-prime
Hence the contradiction

CASE 3:
 n mod3 = 2
then (n+4) mod3 = 0
this makes number n+4 non-prime
Hence the contradiction

In all the cases we can see that there exist atleast one of the 3 numbers n, n+2, n+4 to be non-prime (3 as a factor).
Hence our assumption is false.


 

Answer 2

Using Direct Proof :

Every integer n can be written in one and only one of the following forms: n=3k OR n=3k+1 OR n=3k+2 , where k is some integer.

Case 1: If n=3k, then n is not prime because 3 is a factor of n and n>3.

Case 2: If n=3k+1, then n+2 = 3(k+1) is not prime because 3 is a factor of n and n>3.

Case 3: If n=3k+2, then n+4 = 3(k+2) is not prime because 3 is a factor of n and n>3.

So, we can’t represent n in any of the 3 forms. Therefore there does not exist an integer n > 3 such that n, n+2, n+4 are each prime.