Mersenne primes and BQD Iterator

Using the BQD Iterator it's possible to generate an arbitrary number of binary quadratic Diophantine equations, including ones related to Mersenne primes.

Consider that given:

u² + (2^c - 1)v² = F

then it must also be true that

(u - (2^c - 1)v)² + (2^c - 1)(u + v)² = 2^c*F

And here an adjustment can be made if u and v are odd, as then 4 will automatically be a factor that can be divided off, so the adjusted BQD Iterator is:

Given odd u and v with:

u² + (2^c - 1)v² = F

then it must also be true that

[(u - (2^c - 1)v)/2]² + (2^c - 1)[(u + v)/2]² = 2^c-2*F

To get to Mersenne primes the iterator is started with: u = v = 1, or u = v = -1.

Which gets you at the start F = 2^c.

And an infinite number of iterations are available.

The first iteration, with the adjusted BQD iterator, and u = v = 1, or u = v = -1, is:

[(1 - (2^c - 1))/2]² + (2^c - 1) = 2^2c-2

Next iteration:

[(1 - (2^c - 1))/2 - (2^c - 1)]² + (2^c - 1)[(1 - (2^c - 1))/2 + 1)² = 2^3c-4

With that 2^c in there it's a lot harder to get a sense of it, so will try some examples. I know that 7 is a Mersenne prime, so will show c=3, with u = v = 1. Then the start is:

1² + 7*(1)² = 2³

Next iteration: (-3)² + 7 = 2⁴

And next is: (-5)² + 7(-1)² = 2⁵

And another iteration gives: (1)² + 7(-3)² = 2⁶

One more gives: (11)² + 7(-1)² = 2⁷

This example is interesting as 7 is left bare for several cases which is something that was noticed by Ramanujan. Let's move on now to 15, which is interesting for not being prime. Curious to see if there is some indication by this approach.

1² + 15*(1)² = 2⁴

Next iteration: (-7)² + 15 = 2⁶

And next is: (-11)² + 15(-3)² = 2⁸

But I'll note you can also switch signs at these iterations, but you can't always switch signs as if u+v = 0 then you get no further iterations, but here it's ok, so let's use 7 instead of -7.

Then the next is: (-4)² + 15(4)² = 2⁸

That gives a factor of 16 which when divided off just drops back to the earlier result. Interesting.

Now each of those examples can be used for factorizations, where the first non-trivially factors 15, the second does not, and that last does.

So it's not like the mathematics cares here to hide whether or not 2^c - 1 is prime. It's not clear to me though if it's also giving a reason for when it would have to be composite though I suspect one may be in there. Of course it'd be more fun to me if there is. Doesn't make it so.

That mathematical intuition thing fascinates me.

The iterator goes out to infinity. That's a much bigger deal for a prime. Having to fulfill the conditions for the iterator is an infinite burden which the math handles of course. But I suspect it's a LOT easier to handle that infinity with a composite than with a prime.

Which is an attempt to try and get a handle. It's interesting you can look at an area where so many people have looked before, and see it in a different way. But of course, mathematics is an infinite subject. Somehow you can figure things out--though it may take a while.

Oh well, that's enough for a start to an investigation. Wanted to step through some early thoughts after seeing that the BQD Iterator could be used with Mersenne primes. Wondering if it can explain some things about them. It may take years but I suspect there is an answer in there, somewhere.

It's interesting that Ramanujan helped me in this direction. I've had what I now call the Binary Quadratic iterator for over 6 years. It was just one more of those things I have laying around.

I had used it for things like explaining the fundamental solution to x² - Dy² = 1. But a few days ago I decided I needed to name the thing. And after naming it, I decided to try something.

I wonder why naming things can have such power on the imagination.

Oh well.

Whatever works, I like to say. Definitely have had fun.

And learned a few things. And that works for me.


James Harris