Questions about convergence testing.

[Rudy Toody]

I have found an interesting infinite sum that appears to converge to a number.

In fact, I have created four of these using slightly different rules.

Since, all of the odd primes are involved in each series, I know that the series are infinite. I will confirm this by using Euler's Prime Product in place of the sum at each step. I should get Pi^2 / 6.

Now, I have one more series that is nearly identical to one of the above, but it drops an occasional prime (about one prime for every 8 steps.) The two series are in lock-step for 7 out of 8 steps.

That series also seems to converge to a number.

Question1: If I can prove convergence of the series that contains all of the primes, and if I can prove that when using the ratio test that they converge, would that indicate that the second series is also infinite? I know it would mean that it is convergent.

Question2: Can I use the Prime Product as one (or both) elements of the ratio test, or does that only work for sums?

 

[AlephZero]

This is a math forum. if you want some feedback on your ideas, you will have to write down some math.

There are some excellent mathematicians here, but not many experts at ESP.

 

[Rudy Toody]

This is a math forum. if you want some feedback on your ideas, you will have to write down some math.

Think of it as a story problem.

Edit: I think this should be in the Number Theory forum. Perhaps, a moderator could move it.

 

[Rudy Toody]

I have found that these series sum to a numbers whose convergence/divergence is undecidable.

Using this information and a few more rules, I have created proofs of Andrica's Conjecture, Legendre's Conjecture, and using a slightly different method, Goldbach's Conjecture.

It seems that the Twin Primes and Quadruplet Primes might also be targets for this series.

I would not have found these calculations had I not been ignored by this forum. Thanks for the tough-love!

I am drafting the proofs now and will post again after I submit them to a journal.

 

[HallsofIvy]

So far you have made three posts and said you have some series and want to know if it converges. That is a lot like saying "I have this problem, and want to know the answer"! There simply isn't enough information to tell us what your question reall is!

 

[Rudy Toody]

So far you have made three posts and said you have some series and want to know if it converges. That is a lot like saying "I have this problem, and want to know the answer"! There simply isn't enough information to tell us what your question reall is!

In my previous post, I stated that I had found the answer. I am in the process of drafting the proofs.

It turns out that these series are entirely new and magical.

Bounded, Convergent Prime Series Abstract:
1) We create a new type of infinite prime series using bounded steps to force convergence. 2) Using one of these series, we show a proof of Andrica's Conjecture. 3) Using that proof, we show a proof of Legendre's Conjecture. 4) Using that same series in a slightly different manner, we show a proof of Goldbach's Conjecture. 5) Using another series and the series from 4, we show a proof of Goldbach's Weak Conjecture.

 

[Rudy Toody]

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"

http://math.rudytoody.us/BisectedPronicPrimeSeries.pdf" It was rejected for obvious reasons. You can use it to get a sneak-peek at the proofs I will present on the other thread to see how magical this Semi-Pronic series can be.

If anyone has suggestions, feel free to jump onto that thread.

If you come up with your own proof, publish it! (with a citation to give me some credit.)

Thanks.