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From: Arte Atem on 19 Jul 2010 18:33
Recently an idea I had several years ago about prime numbers seemed to
me to be actually an axiom about prime numbers themselves, which was
rather exciting and I suggest that it is surprising that the world has
either yawned or is not aware of how significant it is, but it among
other things quite simply explains the why of prime gaps. In this
post I'll quickly show how that is the case by focusing on the twin
primes.

An example of twin primes is: 11, 13

The gap between them is exactly 2, and one way of looking at "why" is
to note that if for any odd prime p less than sqrt(11), if (11+2) mod
p is 0, then the prime gap can't occur. May seem trivial but it is
the key to understanding the twin prime gap.

For instance look at 13, where the next prime is 17. That's because
(13+2) mod 3 = 0.

That's it. It's the only reason. Mathematically THERE IS NO OTHER.

But also notice it means that 13 mod 3 = 1 is needed. So if 13 didn't
allow itself to have 1 as a residue modulo lesser primes then that
would have been a second twin primes case, but that is ludicrous. How
could the prime 13 decide that it doesn't like a particular residue
mod 3?

And that's your clue. If you understand that one thing then you can
grasp the why of twin primes and then of arbitrary even prime gaps, as
for instance for that gap of 4 between 13 and 17, you needed (13+4)
mod p, where p is an odd prime less than sqrt(13) to not be 0, for,
once again 3.

Mathematics doesn't need anything else to say a prime gap is there!
There is ONLY one way to get a prime gap, which is for

(p_1 + g) mod p_2

to NOT be 0 for any odd primes less than sqrt(p_1). If it is, then
that gap does NOT occur.

So the requirement is that p_1 NOT equal -g mod p_2, for all primes
p_2 less than sqrt(p_1).

If no residues are excluded or preferred by the primes then there will
the Twin Primes Conjectualways be cases where those conditions are met,
which trivially proves
re.

(Think carefully and now you can disprove Goldbach's Conjecture
trivially as well, but without a counterexample likely to ever be
directly seen which is unsatisfying I admit.)

Another way of looking at it is, if you're looking at bigger and
bigger primes p_1, and you are getting all these residues modulo
primes less than sqrt(p_1), and there is no preference by the primes
then just at random at times you will have cases where -g is not a
residue modulo ANY of those primes which will give you a prime gap.

(Primes may here define random for the real world.)

So those go out to infinity.

What's interesting about this issue may be that people rarely talk
about the why of prime gaps so it seems like a hard problem,
especially if you wrap the prime distribution into it which you can do
with a false correlation. That is, as the count of primes drops as
you get bigger numbers, necessarily the count of twin primes will drop
as well, but the prime distribution itself is irrelevant as to the
reason of the "why" of twin primes or other prime gaps.


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