Tuesday, December 09, 2014

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