[tie_]

Isn't 4k+1/4k-1 rule trivial though?

If we take any number K=N*4 divisible by 4 and >2, that'd be an even number by definition. The two closest odd numbers on either side would be (K-3), (K-1), (K+1), (K+3). As it happens (K+3) is the same as K(-1) for the next N, and (K-3) is the same as (K+1) for the previous N. So _all_ odd numbers follow this rule.

What "4k+1 or 4k-1" says in a roundabout way is that all prime numbers (>2) are odd, which isn't much of a surprise.

[naniwaduni]

So is the 6k±1 rule: 6k and 6k±2 are all even, 6k±3 is divisible by 3. You can extend this further: all primes greater than 5 must take one of the forms 30k±1, 30k±7, 30k±11, 30k±13. This is much less exciting, but ... suggestive. (No, not that suggestion, that one isn't actually true.)

For a certain point of view, most of math is trivial corollaries.

(Proof: check.)

[BadOakOx]

I think this still seems trivial with a 5th grade algebra knowledge.

Additionally, for the 4k+1 / 4k-1 topic, it is just a complicated way of saying 2k+1 (as parent suggested).