- #1
Paul Mackenzie said:Hi All;
I attach a pdf file on something I have been working on for some time.
Any feedback would be appreciated.
Regards
Garbagebin
Norwegian said:Hi, I found the pictures at the end very nice and interesting. I have never seen this Goldbach comet before, and your scaled version following it shows even clearer the "good" values of n, and the structural bands of the comet.
(The goldbach comet plots the number of prime pairs adding to n, while the 2nd picture plots the ratio of primes belonging to such a pair.)
I presume those n having many small different prime factors will give a high valued point, so the "local max" around 30k must be the product of primes up to 13. Then I guess that second floor of points are for those N divisible by 6, and the bottom floor those who are not. Looking at this comet of course begs the question, how come nobody has yet proven Goldbach?
Back to the paper: should be more structured, with an introduction including a summary. Your mathematical language is not very precise, and you will lose many readers there. Then, the statement that zero is not even, was surprising, but I continued hoping for a "proof" of goldbach or something. That was not to come, but instead a (de)tour into Fourier analysis, where a statement about the symmetric properties of a sequence were translated into a statement about real/imaginary parts of a transformed sequence. No indication as to why this would be useful in relation to the rest of your paper was given. Instead we jump to a different and imho more fun subject, prime numbers and goldbach partitions, and you give us your nice pictures.
Paul Mackenzie said:I have some source code which computes the number of goldbach partitions from the Fourier transform of the prime number sequence. If anyone wants a copy please do ask.
SteveL27 said:Can you back up to the beginning and help me to understand your notation?
What's the sequence for the even numbers? What's the sequence for the primes? I think if you can simply show a couple of examples, we can help sort out the notational issues.
It's not possible for anyone (well, for me, anyway) to follow your argument since the definition of f[x] was so garbled. I get that to each term of a sequence you assign a pair of numbers, but I'm unclear on what those numbers are. Then you start talking about 2N, sort of out of the blue, without defining N. And if you're not sure whether 0 is an even number, let's talk about that.
In other words, let's nail down the Intro before going forward.
Paul Mackenzie said:Hi Steve and all:
Thanks for your help regarding the notation.
To understand where I coming from, I will describe f[x]
with reference to the prime number sequence {2,3,5,7,11,13}.
In this example the sequence is limited to all primes less
than 16 (viz 2N = 16).
I attach a pdf file which shows the functions f(x), f(2N-x),
and f(x).f(2N-x) where 2N = 16 and
where f(x) = 1 when x is prime
SteveL27 said:I'm just asking you to define f clearly. It shouldn't take more than two or three lines. Tell me its domain, tell me its range, tell me what it does to a typical input value. I don't know how else to say this. Just tell us what f is. Two or three short lines. All the rest of this exposition is running before you've caught the ball. Just define f. That's step 1. Beyond that I can't offer any more advice.
For the purposes of this post, a sequence of integers is represented in the x-y Cartesian coordinate
system in the following form.
f [x]=1 if x is a member of the sequence, otherwise
f [x]= 0 , for all x = 0,1,…, 2N.
Norwegian said:But I hope further comments can be about the contents of the remaining paper, and not just from people pretending not to understand its first page.
The symmetry of a finite sequence of numbers refers to the property of the sequence being unchanged when it is reversed. This means that the sequence reads the same forwards and backwards.
To determine if a finite sequence of numbers is symmetrical, you can simply reverse the sequence and see if it is identical to the original sequence. If it is, then the sequence is symmetrical. You can also check if the first and last numbers are the same, the second and second to last numbers are the same, and so on until you reach the middle of the sequence.
Yes, a sequence of numbers can be partially symmetrical. This means that only a portion of the sequence is unchanged when reversed, while the rest of the sequence is not symmetrical.
No, not all finite sequences of numbers are symmetrical. It is possible for a sequence to have no symmetry at all, or to have partial symmetry as mentioned in the previous answer. It is also possible for a sequence to have multiple symmetrical portions, making it symmetrical in multiple ways.
Symmetry of a finite sequence of numbers is useful in mathematics as it can help identify patterns and relationships between numbers. It is also commonly used in statistics to determine if a data set is symmetrical or skewed, which can provide valuable insights in data analysis. Additionally, symmetry is a fundamental concept in group theory, which has applications in various fields such as physics, chemistry, and computer science.