Recent content by Paul Mackenzie

I am able to take this matter a further step forward on this topic. The whole aim of this exercise was to find relationships between the number of goldbach partitions g[2N-2], g[2N], and g[2N+2] for arbitrary consecutive even numbers 2N-2,2N, and 2N+2. and THEN somehow argue it is not possible...

I just noticed I made a typo in my previous post of Jun3-12. The statement "Thus if we multiply the LHS and RHS of this sum by e ^{-j2π2/2N} all these components will go to zero with the exception NOW of g(2N-2) and k(4N-2)" contains a typo. It should read "Thus if we multiply the...

Hi Haruspex: I am counting the number of consecutive runs of similar digits in the sequence. So the run x1x refers to either the subsequence 0,1,0, or 0,1,-1 or -1,1,0 or -1,1,-1. As another example the consecutive run of two positive ones viz x,1,1,x refers to the subsequences -1,1,1,0...

I have been investigating goldbach partitions for some time. One interesting observation I have been able to determine is concerning the "direction" of the goldbach partitions whether they are increasing or decreasing as 2N increases. To get an idea of this I constructed a function f(2N)...

I have been able to take the aforementioned post one step forward. As mentioned earlier I was able to show that the number of goldbach partitions g(2N) for some even number 2N was equal to g(2N) = \frac{1}{2N} \sum^{l=2N-1}_{l=0}F2[l] where F[l] is the Fourier transform of the prime...

I show below how I arrived at the relationships I mentioned in my previous post. My Apologies for not doing it before. Firstly, I considered the properties in the Fourier domain of the following function. f(x) = 1 if x is prime f(x) = 0 if x is otherwise for all real x. In particular...

I found the following relationship concerning goldbach's conjecture; viz that every even number is the sum of two primes. If goldbach's conjecture is true then the following must hold for all 2N \sum^{2N-1}_{l=0} ( \sum^{p < 2N-1}_{ p odd primes=3} cos (2πpl/2N) ])2 >...

Hi All; The following attachment shows a diagram of the ratio R[2m] = g^2[2m]/g[2m-2]*g[2m+2] where g[2m] is the number of goldbach partitions for the even number 2m. What is the reason for the "forbidden zones". I understand this is somehow to do with the factors of the even number...

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...

Thank you and steve for your comments. I made a big error concerning zero and even numbers, and have corrected that. As you said the notation concerning f[x] needs some clarification which I will attend. I will add a more thorough introduction and conclusion. I was not really trying...

Hi All; I attach a pdf file on something I have been working on for some time. Any feedback would be appreciated. Regards Garbagebin

Hi I think I have found an upper bound for C[2N] [though it does not hold for some small 2N]. Again an intuitive guess. C[2N] < 4*Pi[2N]*Pi[N] - 2*Pi[N]*Pi[N] - P[2N]*Pi[2N] so we have Pi[N]*Pi[2N] < C[2N] < 4*Pi[2N]*Pi[N] - 2*Pi[N]*Pi[N] - P[2N]*Pi[2N] I checked this for odd primes upto...

To better understand the terminology I have included a word document with a graph with values 2N= 6 to 72,000. It appears the term R[2N] you referred to and G[2N] are interchangeable. Regards Paul

I have been able to prove a weak version of the above. Let the cumulative sum C[2N] = Sum([G[2N]; 2N= 6 to 2N) where G[2N] is the number of goldbach partitions for the even number 2N. Consider every combination of odd primes less than N, note that the sum of these combination pairs is less...

I noted the following concerning the cumulative sum of Goldbach partitions C[2N] = sum[ G(2N) ;from 6 to 2N] is greater than pi[2N]*(pi[2N] -1)/2 where 2N is an even number 2N=6,,,,, C[2N] is the cumulative sum of the goldbach partitions of the even numbers 6,...2N G(2N) is the...