**1.**

*To what extent does what I call the Binary Quadratic Diophantine iterator explain the behavior of Mersenne numbers?*

It is quite well-known that Mersenne primes are rather rare, and considerable effort is being expended in finding them. But can we use the BQD iterator to explain why they are so rare?

**2.**

*How well does what I call the Prime Residue Axiom predict the behavior of actual numbers?*

The PRA examples I give are with twin primes, but if it is an axiom then it gives predictions for indefinite sized gaps. While such evidence might not be convincing for those doubting the axiomatic quality they would definitely be compelling. While a predictive failure would force me to answer or accept that the notion fails.

**3.**

*Does numeric integration of the partial differential equation that follows from my prime counting function agree with the predictions of other mathematics?*

The prime counting function I found is distinctive in that it recurses, and counts primes by calling itself.

That means it leads to a difference equation which is still Diophantine. But that leads to a partial differential equation which is not. So yeah, I do have research which is not just about integers! This one is good to leave as the last as it relates to the Riemann Hypothesis which I tend to very deliberately avoid. To me RH is just too emotional an area for mathematicians so I don't think they're rational about it. And I'm not a mathematician so I don't really care about it anyway. So I leave everything open for others as I think it worthless for me to bother.

Gave links I think related to the questions in each though there are more blog posts around each. I have fiddled around with lots of things as the spirit moved me.

And those are some open research areas.

And that's good for a start! Just had an impulse to write a post like this one and see how it goes, so quickly came up with some things that occur to me.

I actually have quite a lot of open research areas. May as well talk some of them out. These are three that might intrigue others, I'm guessing.

If this way of doing things works well maybe I'll toss some others up.

James Harris