I can prove that the mapping of the two-step series mapped to the multiple-step series always returns the same values. However, if my series returned a 1 in one of its steps it will not change this equivalence because each sub-step is part of a product (both series.)
Another reason for the demanding proof is this Riemann identity I discovered by accident:
\text{Zeta}[3]==\frac{2 \pi ^4}{315\prod _{n=1}^{\infty } \frac{1}{1+ \frac{1}{\text{Prime}[n]^2-\text{Prime}[n]}} }
Sorry about the red never, ever. I need to prove it is impossible the have a missing prime in one of my series' segments. Just showing it to millions of steps is not good enough for the number theorists.
Incidentally, the other series is based on Betrand's Postulate, where it has been proved...
Yes, it would diverge. That is my dilemma. I can take the series up into millions of steps without encountering a 1. The series can be shown to converge because this exception never happens. I need to prove that it will never happen.
The question I'm trying to answer is: If both series use essentially the same method at each step to identify primes, and one series has been proved to never return a 1 (which indicates that no primes were found at that step), can I use the fact that the second series, which can be kept in...
Ok, a_n and b_n will never equal 1, so that takes care of two out of three conditions.
Now, I seem to remember a mention somewhere about a possible side-effect in a convergent sum. That if the sum converges to some point without the side-effect occurring, that the side-effect would not occur...
I have a convergent sum where I use the reciprocal of a_n at each step:
a_n = a_n / gcd(a_n, b_n) <--- I'm removing common factors.
This converges as long as I want to run it. Both a_n and b_n are quite dynamic.
However, if a_n equals b_n then the divide after the gcd would return 1...
I have discovered that the series I found is actually Opperman's Conjecture. (1882) And it is stated that if it is proved, the others I mentioned will also be proved.
So, if we can prove that the series is infinite because an exception (missing prime within a gap) will never occur, then we...
The important theorem would be the one that proves the function in the first link. I'm trying to show how important that function is. It is magical. It can be used for additional proofs, too.
If this series https://www.physicsforums.com/showthread.php?t=485665 is proved to be infinite, then proofs of these two conjectures can be done as simple corollaries.
Legendre's Conjecture states that for every $n\ge 1,$ there is always at least one prime \textit{p} such that $n^2 < p <...
First, I want to apologize about my curtness earlier. I did not have any math at that time to show. Now I do. I have started a thread on the Number Theory forum to discuss this series.
Edit: https://www.physicsforums.com/showthread.php?t=485665"...
A while back I posted a question about this series on the General Math forum and was brought to task for not showing any math.
My hope is to prove that these series are infinite.
http://oeis.org/A002378" are the series 0,2,6,12,20,30... and distances between consecutive numbers are increasing...