Question

What is proof for statement that every prime number is either of form (6k - 1) or (6k + 1) ? But, reverse is not true.

Answer 1

Any number can be written in the form of (6k+r) ; where r ∈ {0,1,2,3,4,5}
Among these only (6k+1) and (6k+5) cant be factorized (without knowing the value of k)
Hence, every prime number can be written in the form (6k+1) and (6k+5) ≡ (6k-1)

The converse is not true as a number of type (6k±1) can be factorized for some values of k (try to find an example).

Note that: 2,3 violates this rule (try to find why?)

Answer 2

What does this mean (6k+5) ≡ (6k-1) ??