Re: Pattern to the primes , intresting universality.

quasi wrote:

On Mon, 18 Feb 2008 11:18:16 -0800 (PST), Rob Barry
<rob.barry@xxxxxxxxx> wrote:

On Feb 18, 8:18 am, quasi <qu...@xxxxxxxx> wrote:
On Sun, 17 Feb 2008 20:02:58 -0800 (PST), Rob
Barry

<rob.ba...@xxxxxxxxx> wrote:
Let F(x) = (cos(pi * gamma(x) / x) + cos(pi / x))
^ 2 + sin(pi * x) ^
2

Then, for all x > 1, F(x) = 0 iff x is a prime
integer.

Since the group seems to have made a bit of
headway in accepting this
fact -- or at least the possibility -- my
question is simply this: Why
is this not the "pattern" (or one of many such
equivalent patterns)
that underly the distribution of the prime
numbers?

Sure, it's one of many such equivalent patterns.

Essentially it's an exact equivalent of the proved
pattern (.i.e.
theorem)

(n - 1)! = -1 (mod n) iff n is prime

There are other "patterns" for the primes. For
example,

p|ab => p|a or p|b

As one more example, a finite ring with 1 is a
field iff it has a
prime number of elements.

Thus, the primes are anything but random.

Truly random integers wouldn't satisfy laws such
as the above.

Again, I understand that it can't be the pattern,
because my training
in math is certainly not advanced enough -- a
simple BS in the subject
-- to aspire to such mathematical heights. I
simply want to know why
it fails to define a pattern

It does define a pattern -- but it's a _known_
pattern -- the
Wilson's Theorem pattern. Converting it to
trigonometric form and/or
displaying it graphically doesn't make the
underlying pattern any
different.

in what I've always heard was the essentially
"random" distribution
of prime numbers.

What you "heard" was an oversimplification. There
are many things
about the pattern of the primes that are as yet
unknown. But at the
same time, we also know a lot about them. We have
_theorems_ (laws)
which the primes (would-be anarchists that they
are) have reluctantly
agreed to satisfy.

I see. This answers a lot. Thanks for taking the
time to explain,
though I must admit I'm still a bit befuddled as to
why, if one can
construction an oscillating function whose domain is
the reals and
whose zeros are exactly the prime numbers, one has
not, in essence,
defined a pattern for their distribution.

It _does_ define a pattern.

It's just that's it's not a _new_ pattern -- just a
reformulation of a
known one (Wilson's Theorem). As to whether the
reformulation in terms
of a continuous function can yield any new insights
-- it's possible,
but not likely.

The main misconception on your part is your assertion
(based on what
you "heard") that the set of primes has no pattern,
no structure.
That's false.

The very definition of prime contradicts any claim of
"randomness":

If p in N, p > 1, then p is prime iff x in N and x|p
=> x = 1 or x =
p.

Would a "random" set of positive integers satisfy the
above condition?
Of course, not.

But I suppose the explanation is just going to be
over my head,

It's not really over your head. You just didn't get
the answer you
were hoping for.

so I'll go away.

No need to go away. Even if your function appears
unlikely to provide
new insights into the structure of the set of primes,
it was still of
interest, if only to see the tie to Wilson's Theorem.

quasi

yes , and also intresting might be the universality.

meaning that this formula can be extended to z.

and if f(z) has the primes as zero on R.

then f(z) = q(z)* (function mentioned here (z) )

( f(z) and q(z) analytic )

and who knows , we might be able to derive at a RH- like theorem for f(z) , perhaps even provable.

now your function is probably considered by others before and other functions have similar properties , but this universality and perhaps even possible RH-like theorem make it intresting.

im sure it has been investigated before.

but i also guess not deeply enough.

disclamer : perhaps a simple and direct transformation to the zeta-function is possible and makes the idea not intresting afterall.

reducing to the many essays on RH.

but i doubt that.

zeta's zero's relate to primes , but the primes dont give zero's directly.

so feel free to stay.

its rare that we get a nice idea within the first 15 posts anyways.

regards
tommy1729
.

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