Question

Show that any prime number can be expressed in the form of 6k +/- 1 (k can only be natural number) for number greater than 3.

Answer 1

All integers can be represented as 6k+m, where m ε {0, 1, 2, 3, 4, 5}, and k is some integer. This is obvious. Therefore:

m=0: 6k is divisible by 6. Not prime 
m=1: 6k+1 has no immediate factors. May be prime. 
m=2: 6k+2 = 2 x (3k+1). Not prime 
m=3: 6k+3 = 3 x (2k+1). Not prime 
m=4: 6k+4 = 2 x (3k+2). Not prime 
m=5: 6k+5 has no immediate factors. May be prime.

Therefore the only candidates for primacy are 6k+1 and 6k+5.
6k+5 = 6m-1 for m=k+1.
Therefore, all primes are of the form 6n±1 for some integer n.   ( Hence proved )