Summing squares to powers

Sometimes I just like to play with numbers:

1² + 9*1² = 10

8² + 9*2² = 10²

26² + 9*6² = 10³

28² + 9*32² = 10⁴

Which goes out to infinity, where I used a simple rule.

If: u² + 9v² = 10^a

Then: (u - 9v)² + 9(u + v)² = 10^a+1

Which is just using my BQD Iterator. And notice that u and v can be positive or negative, while I like to show positive as is easier and looks prettier, which allows some selectivity, which I used behind the scenes to get my series above. Some choices made it more boring.

Like yeah so if you use a = 4, u = 28, and v = -32, then next is:

316² + 9*4² = 10⁵

So yeah, every power of 10 can be written as a sum of two squares.

And was fascinated a couple of years ago that the general result is, for an integer n equal to 0 or higher, and an integer m equal to 3 or higher:

x² + (m-1)y² = mⁿ⁺¹

I have m raised to n+1 so that n is a count of iterations. And if m-1 is a square then every power of m can be shown as the sum of two squares.

And talk it all out in this post. So I just used m = 10 above.


James Harris