Doing some iterations with BQD Iterator

Decided to check something yesterday using what I call the BQD Iterator, and was thrilled at intriguing results. Here I'll focus a little more closely on one of them.

Yesterday I found the following:

x² + 2y² = 3ⁿ⁺²

must have integer non-zero solutions for x and y, for n = 0 and any positive integer n.

The iterator here is:

u² + 2v² = F

then it must also be true that

(u-2v)² + 2(u+v)² = 3F

One starting result for x and y, for n=0, is u = 1 and u = 2:

1² + 2*2² = 9

For n = 1, iterator gives next solution is x = -3, y = 3:

3² + 2*3² = 27

For n = 2, next is x = 3, y = 0. So that stops. Interesting.

Here's another option.

Starting for n=0, with x = 1, and y=-2:

1² + 2*(-2)² = 9

For n = 1, iterator says next is x = 5, y = -1:

5² + 2*(-1)² = 27

For n = 2, next is x = 7, y = 4:

7² + 2*4² = 81

For n = 3, next is x = 1, y = 11:

1² + 2*11² = 243

That's more interesting for a bit, but next iteration with positives will pull in factors of 3 again. Which can be avoided by using a negative value again.

I'll do that just to see, using -11 for y.

For n = 4, with that change iterator says next is x = 23, y = -10:

23² + 2*(-10)² = 729

So you can just switch to a negative whenever to avoid having 3 as a factor of x or y.

Interesting but tedious.

So it can be a little tricky using the iterator.


James Harris