Re: Towards a Formula for Primes

On 7 Mai, 00:33, Gerry Myerson <g...@xxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

.. "charlesweh...@xxxxxxxxxxx" <charlesweh...@xxxxxxxxxxx> wrote:
.. > Perhaps within this concept lurks the narrow gateway - the key -
to
.. > entering into the solving of the problem of finding a test of, or
.. > formula for, primes.
..
.. The problem of finding a test for primes was solved by Euclid.

You do not appear to be very rigorous in your contemplation of what I
have said. Of course there are such things as the "Sieve of
Eratosthenes", which laboriously go along a number-line in jumps of
two, three and so on until all non-primes have been struck out.

The thread I opened here is titled "TOWARDS a Formula for Primes". It
is based on what is very likely to be a new discovery, but I have not
explored it in enormous depth because I have other things to do.

.. The problem of finding an *efficient* test for primes, well,
.. that one is still open

THAT is what it is all about. It is about the possibility of creating
a formula that can be applied to GENERATE primes, or a formula that
will test a number to see if it is prime. It is not a suggestion for
another "Sieve" procedure.
..
- but I'd be more confident of your chances
.. if I thought you knew anything about the progress that has been
.. made on that problem over the last two thousand years, and
.. especially over the last 25 years or so. Are you familiar with,
.. say, the Adleman-Pomerance-Rumely-Cohen-Lenstra method?

That is a very glib loaded question. What I know for sure is that YOU
YOURSELF do not know all the research that has been done in the last
two thousand years. Until you have made YOURSELF familiar with ALL the
research of the entire PLANET over the last 2000 YEARS you cannot
condemn others who have not made that effort.

.. Do you know how good it is? Do you have any reason to believe
.. that your concept will lead to anything better?

Lenstra comes through in my memory in connection with something I was
doing which overlapped. This name rings a bell, and I am sure the work
is good. I am not trying to become the "PRIME-FINDING CHAMPEEN OF THE
WORLD". This is not a prize fight.

I am familiar with the nonsense work of somebody whom I will call
"Herbert Herbert". He was a journalist, not a mathematician. He wrote
a book in which he gave his "Formula for Primes". It was very simple
and delivered the first seven.

At the same time I was reading "The Art of Computer Programming" by
Professor Donald E. Knuth. In one volume, he gave a formula for primes
which worked only for the first seven. He warned that you have to try
such formulae for the eighth and more, because they may not work.

"Herbert Herbert" was hailed in the Guardian newspaper as the
discoverer of the long-sought efficient formula for Primes. The
Japanese beat a path to his door, and he was invited to Japan. He was
feted in the universities, and introduced to Japan's chief economist.

Meanwhile, the eminent crystallographer Alan Coutanceau-Clark, the
world authority on the geometry of nested spheres, and a very dear
friend, examined the book and found that the formula did not deliver
the eighth.

I had written machine-code to illustrate one of Alan's theorems. I
used the obvious formula 6n plus-or-minus 1. It was quick enough, but
does not deliver the 2 or 3. In addition, it delivers false primes. 25
is 6*4 plus 1. It is non-prime. Similarly, 35 is 6*6 minus 1. It is
non-prime. Similarly, 49 is 6*8 plus 1. It is also non-prime. I simply
used a "Sieve" to test only the output of 6n plus-or-minus 1, and the
"Sieve" decided the outcome. When the algorithm delivers all the
primes (except 2 and 3) it is good, but spoilt by the false primes,
leading to the need for further testing.

"Herbert Herbert" returned from Japan, showing off his photos of
himself with professors and economists in japan. We were chuckling.
Then, he was summoned back to Japan. It seems the Japanese, having now
overcome the language barrier, had tested his formula for the eighth.
He was required to explain himself. Why, after all that fuss, did his
formula not work?

What is so special about the Pythagorean Perimeters Theorem is that it
can convert a coprime DUPLE to a coprime TRIPLE. Therefore, built into
it is the concept "NEW PRIME". It requires some careful and rigorous
analysis to find out the exact rules as to what kind of "NEW PRIME" is
created.

EXAMPLE:
the 3:2 rectangle delivers 8:15:17

The 2 is delivered three times in the eight.
The 3 is delivered once in the 15.

The 15 also delivers NEW PRIME 5.
The 17 is also a NEW PRIME.

So we get two new primes, but I only guarantee one. That is because
when a duple becomes a triple, only one is added. The other new prime
is spurious, and cannot be guaranteed.

The 3:1 rectangle delivers 7:24:25

The 7 is a NEW PRIME.
The 3 is contained in the 24.

The 24 also delivers NEW PRIME 2 (three times)
The 25 delivers NEW PRIME 5 (twice).

Duples can be picked out from the triple, and so yet more NEW PRIMES
will appear. There will also be some redundancy. Accordingly, there is
room here for research - to maintain the concept of "NEW PRIME" until
the essence of the primality of numbers can be found. That leads to a
new formula.

When Pythagoras produced his theorem, he did not prove that 3*3 plus
4*4 is 5*5. If you have not studied what Pythagoras actually found -
the "Locked to the Rightangle Grid" concept - I suggest you
investigate it. Even a child can multiply three by three, add it to
four by four and compare with five by five. The question was deeper,
but takes too many words to describe.

Similarly, here, I do not make this suggestion lightly. The axioms of
arithmetic are obviously too incomplete for a solution to the problem
of a formula for efficient prime-generation, or for efficient prime-
testing, to come from arithmetic alone.

The Pythagorean Perimeters Theorem is a very, very simple GEOMETRICAL
theorem. There is something in the GEOMETRY of a single Diophantine
rectangle, translating into a single Diophantine right triangle of the
same perimeter that throws up a guaranteed "NEW PRIME".

So there will be no "false primes". However, whether any resulting
formula delivers an array of primes with some primes missing, or
whether it delivers them all, depends on whether the approach can lead
to a formula. Only after investigation of the suggestion can it be
decided whether the approach has its merits.

When the axioms of a system are incomplete, as probably here in
arithmetic, one reaches into another system, as in geometry here. That
is how problems are solved.

Mine is not an ANSWER. Mine is a discovery that can be taken further.
I have found a source of "NEW PRIMES". You work with two primes and
find you have got three.

Charles Douglas Wehner



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