More power 10 sum two squares

From my research is now known how to easily show how to write every power of 10 as the sum of two squares. Which I think is cool. Thought would post more about it. From last iteration given in my previous post on subject I have:

316² + 12² = 10⁵

Where have the rule, from my BQD Iterator.

If:  u² + 9v² = 10^a

Then: (u - 9v)² + 9(u + v)² = 10^a+1

Noticed that 316, which is an approximation of sqrt(10) times 100, and figure is because other square happened to be small, so am curious, about 10⁷ now.

So will do two more iterations. And can use u = 316 or -316, and v = 4 or -4, so will fiddle with things, which I did. Of course just showing what I decided works ok for my purposes.

(352)² +  9(312)² = 10⁶

Which is: 352² + 936² = 10⁶

Next will use u = 352, and v = 312:

(-2456)² + 9(664)² = 10⁷

Which is: 2456² + 1992² = 10⁷

And didn't go way I thought it might. Had to fiddle with signs and deliberately trying to get a small square again, just ended up getting both sides multiplied by 100. Math does not care what I want. It is interesting to see the math adjust to what you TRY to get it to do. That sense of the math is something. Like there is this intelligence, which I guess, yes it is. It is just perfectly logical.

Guessing though if kept going would find cases where with odd exponent for 10, would again see an approximation of square root of 10 in there. Well, curiosity is satisfied, for now.


James Harris