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From: JSH on 17 Jul 2010 23:14
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
always be cases where those conditions are met, which trivially proves
the Twin Primes Conjecture.

(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.


James Harris
From: MichaelW on 17 Jul 2010 22:52
On Sat, 17 Jul 2010 20:14:51 -0700, JSH wrote:

>
> 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.
>
> James Harris

The only part of the post I would take issue with is with this paragraph.

Let's see if I get this right. For any given prime p_1 we can state the
probability that p_1 is half of a pair of primes with a difference of g.
Call this probability j(p_1,g). This value is always greater than zero
and can be calculated by considering modular residues.

So far so good. The issue I have is that saying that there is a chance of
an event happening does not *guarantee* that the event will happen. For
example if a shuffle a deck of cards and look at the top card there is a
1/52 chance that the card is the four of hearts. Say I do this a billion
times. Now the chance of never drawing a four of hearts is astronomically
small (1 in 10^8,433,167 I think) but this not the same as proving that
it *must* happen.

Of course in our case it is complicated by the fact that we are taking
infinite samples. You could argue that over infinite samples an event
with a finite probability *must* occur but I don't think this is the
case. This is getting past my expertise and I would be interested in the
take from others.

In any case the formula j(p_1,g) decreases as p_1 gets larger (roughly in
proportion to 1/ln(p_1) if I recall correctly) so you have the case of
infinite samples (over all primes) with a decreasing probability. In this
case it is plausible to say that for a certain very large p_1 that
(purely by chance) there are no further prime pairs with a gap of g.

It must be said that the consensus in the maths community is that there
are infinite prime pairs (and indeed k-tuples) of any collection of gaps
that do not trivially generate composites (e.g. p, p+2, p+4 must have one
value =0 mod 3 so (3,5,7) is the only valid solution). This consensus
comes from lines of argument pretty much the same as yours. It is a
formal proof that is not yet available.

Your prime residue axiom does not help as it only asserts preference
(defined as the distribution of residues) which is a probabilistic
statement with the same results as above.

Regards, Michael W.

P.S. I have stuck my nose into areas outside my immediate expertise so I
am open to correction here. Be gentle.
From: Joshua Cranmer on 17 Jul 2010 23:48
On 07/17/2010 11:14 PM, JSH wrote:
> 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
> always be cases where those conditions are met, which trivially proves
> the Twin Primes Conjecture.

There are two steps here:
1. If "no residues are excluded or preferred by the primes." You provide
no proof for that, other than a mere reference to intuition. Number
theory is well-known to be nonintuitive, though, so your proof is no
more than a proof by intimidation.
2. "Then there will always be cases where those conditions are met."
Your first statement is ambiguous, but the most obvious meaning is
probably along the lines that the residues are uniformly distributed
among primes. Under that statement, this tells you nothing about the
conditional dependence of the residues. Specifically, it would not admit
the possibility of two primes (or larger set of primes, in a similar
circular dependence) such that all primes have -g mod one or the other.

These are two assertions for which you give no proof other than
"intuition." But prime numbers defy intuition (and patterns, FWIW).
Furthermore, so does probability. Even experienced mathematicians can
have long debates about what the probability of an event is. I would
like to furthermore add that "uniform" is a very vague assertion as
well: one section of my probstat class opened with "proving" that the
probability of an event, clearly assuming a uniform distribution, was
1/2, 1/3, and 1/4.

> 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.

The reason why it's *hard* is that you have to prove properties about
primes. People may suspect its truth, but it's the demonstration thereof
that is the difficult problem. Prime numbers are little wild animals,
and they defy attempts to be caged in. Show your work, do the proofs:
and use ZFC or its ilk to prove it, don't just say "you know, this looks
true to me, so it must be an axiom."

--
Beware of bugs in the above code; I have only proved it correct, not
tried it. -- Donald E. Knuth
From: Nell Fenwick on 18 Jul 2010 00:55

"JSH" <jstevh(a)gmail.com> wrote in message
news:61df37fe-69ed-46c8-857e-f2401db02437(a)u4g2000prn.googlegroups.com...
> 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.

your use of mod is unfortunate for you.

>
> 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.

No other WHAT ?
You use one simplistic case and then jump to some other conclusion.

>
> 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?

See? told you it would get you in trouble.



>
> 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.

again, this is all trivial stuff. Obvious.

>
> 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.


......DUH.!

>
> 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

What does "excluded or preferred" mean?

> then there will
> always be cases where those conditions are met, which trivially proves
> the Twin Primes Conjecture.

nope, cant jump that far. you are handwaving here.

>
> (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.)

red herring

>
> 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.

talk is cheap, but you havent proved anything


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


you still dont know how to use the word "random", cute.

>
> So those go out to infinity.

marching.......

>
> 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.

..................................

>
>
> James Harris


From: David Bernier on 18 Jul 2010 01:32
JSH wrote:
[...]

> 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).
[...]

Sorry. I have a proof secured in a safe that, for some large
constant C > 0 and for the case g = 2:

"For every prime p_1 > C, there exists a prime p_2 with
p_1 == -2 (mod p_2) and with p_2 < sqrt(p_1)."

I suspect that any such C is even larger that Graham's number,
< http://en.wikipedia.org/wiki/Graham's_number > ;
this might concern people like you, but it doesn't
concern me at all.

David Bernier
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