Tuesday, December 13, 2016

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 = 10a

Then: (u - 9v)2 + 9(u + v)2 = 10a+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

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