Re: [PrimeNumbers] A crackpot's first post
- Geoff Fischer wrote:
> Hello. I am a crack pot.Your are not a crackpot (at least not yet). That would require another
attitude and poor responses to more knowledgeable people, or to being ignored.
So far you show healthy humility.
> Hopefully what I've found is new and different.I'm afraid your observations are not new.
The following is very well-known:
1) All primes above 3 are on the form 6N+/-1, because 2 or 3 divides
all other numbers.
2) Multiplying any number of 6N+1 primes and an even number of 6N-1 primes,
will give a 6N+1 number.
3) Multiplying any number of 6N+1 primes and an odd number of 6N-1 primes,
will give a 6N-1 number.
2) and 3) are considered a trivial consequence (by considering modulo 6) of:
1^n = 1, (-1)^n = -1 for odd n, (-1)^n = 1 for even n.
4) If p>1 divides n, then p divides n+p*k for all integer k (e.g. k=6 in
your observations), so n+p*k is not prime (unless it's the prime p).
What you have is basically the Sieve of Eratosthenes restricted
to 6N-1 numbers or 6N+1 numbers.
This is well-known and often used to compute primes faster.
> Euler commented "Mathematicians have tried in vain to this day to discoverQuotes like these (e.g. the use of "order" in this example) are not
> some order in the sequence of prime numbers, and we have reason to believe
> that it is a mystery into which the mind will never penetrate"
mathematically precise enough to have a truth value. They just seem intended
to give a rough impression that frustratingly many things remain unknown.
There are actually also many things which are known, and the number of
known things is increasing, but not enough to satisfy curious mathematicians.
Sorry to put a damper on your enthusiasm, but:
Considering primes in relation to 6N (or M*N for another M) gives a
well-known correspondence to not doing it, so you will not find new
theoretical results by doing it.
But it can speed up certain computations. It's used a lot for that purpose,
usually with M being a primorial (product of the smallest primes).
Jens Kruse Andersen