Prime Numbers - a property rediscovered

I stumbled across an interesting property of prime numbers last year. The numbers 2 and 3 are the only primes that cannot be expressed as a multiple of 6 plus or minus 1. For example, 7 is 6 + 1, 11 is 12 -1, 13 is 12+1 and so on. However, not all numbers of that are a multiple of 6 plus or minus 1 are prime. For example, 25 = 24 + 1 and 25 is not prime. Another example is 35 = 36 - 1 and 35 is not prime.

After a sleepless night, I put together a short proof of all primes greater than 3 being expressed as a multiple of 6 plus 1 or minus 1.

Steps of the proof:

1. Let S represent a multiple of 6. All numbers greater than 3 can be expressed as either S, S+/-1, S+/-2 or S+/-3. (Why not include S+/-4 and S+/-5? An S+4 number such as 10 is already covered by the next S-2. That is, 12-2=10. Likewise for S-4; for example the number 8 or 12-4, is already covered by the previous S, 6 or 8 = 6+2. Again, for the numbers expressed by S+/-5 are covered by next, or previous, S+/-1.)

2. Since all numbers greater than 3 can be expressed as either S, S+/-1, S+/-2 or S+/-3, then so can all prime numbers.

3. Eliminate reasons for S, S+/-2, S+/-3

    Any number that is S (a multiple of 6) is not prime.

    Any number that is S+/-2 is an even number greater than 2 and therefore is not prime.

    S, as a multiple of 6, is also a multiple of 3 so any number that is S+/-3 is also a multiple of 3 and therefore is not prime (note: in the case of 6-3=3, 3 is prime but we've stated this property is for primes greater than 3.)

4. From Step 2 we established all primes can expressed as either S, S+/-1, S+/-2 or S+/-3. From Step 3 we eliminated primes being expressed as S, S+/-2 and S+/-3, so therefore by elimination, all primes greater than 3 must be able to be expressed as S+/-1.

I wondered why I was never taught this before. Was I dozing through this lesson in math? Certainly someone has come across this property before me. I entered "primes as 6n +/-1" and was greeted by the following ego-deflating statement "this is the most re-discovered property of prime numbers." Nearly the same proof is presented by Chris Caldwell, University of Tennessee at Martin. 6n+/-1.

I didn't at all expect my approach to be unique, but it is always very satisfying to prove something on your own and that should always be encouraged.

Update 3-16-2021

One can build up on this property of prime numbers with the fact that the square of all prime numbers above 3 is 1 plus a multiple of 24. For example, 7 square is 49 which equals 1+2x24. Matt Parker proves this additional prime number fact two different ways in this Numberphile YouTube video.