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Monday, June 12, 2017

Summing seven squares to a square

Discovered a simple technique to build sums of as many squares as you want to a square. For example, here is a sum of seven squares to get a square:

349672 + 7522 + 1128+ 1880+ 26322 + 41362 + 48882 = 357212

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My basic result used to get an arbitrary length of sums of squares is that in general there must always exist nonzero x and y, such that for an integer n equal to 1 or higher, and an integer m equal to 3 or higher:

x2 + (m-1)y2 = mn

Where n starts at 1. Often I like to start it at zero so it's a count of iterations but is prettier starting at 1. And you get values for x and y using what I like to call a BQD Iterator, which is short for binary quadratic Diophantine iterator. I've talked about it a lot.

For a sum of c+1 squares: m = s12 +...+sc2 + 1

x2 + (s12 +...+sc2)y2 = mn

The BQD Iterator is:

Given nonzero integers u and v with

u2 + (s12 +...+ sc2)v2 = F

then it must also be true that

(u - (s12 +..+sc2)v)2 + (s12 +...+sc2)(u + v)2 = (s12 +...+ sc2 + 1)*F

So for 7 squares, I'll need 6 s's and I'll use primes: 2, 3, 5, 7, 11 and 13

Then m = 4 + 9 + 25 + 49 + 121 + 169 + 1 = 378

12 + 377*12 = 378

First iteration: (-376)2 + 377*(2)2 = 3782

Second iteration: (-1130)2 + 377*(-374)2 = 3783

Third iteration: (139868)2 + 377*(-1504)2 = 3784

Which is: (139868)2 + 4*(-1504)2 + 9*(-1504)+ 25*(-1504)+ 49*(-1504)2 + 121*(-1504)2 + 169*(-1504)2 = 3784

Can divide both sides by 16, and get rid of negatives to get:

Which is: 349672 + 7522 + 9*(376)+ 25*(376)+ 49*(376)2 + 121*(376)2 + 169*(376)2 = 1894

And now get final result where will show as all squares:

349672 + 7522 + 1128+ 1880+ 26322 + 41362 + 48882 = 357212

Which is interesting to me, I think. Looks more impressive that way. Of course watch it get built maybe less impressive? But still is, a sum of seven squares to get a square.

Don't really see a practical use, so to me? Is just pure math.

Am curious, if you know a number theorist, why not ask that person to produce an example of sums of squares to a square? It's not like it's actually hard to do, if you know how.

Of course, highlighting a cool result with mathematical tools I pioneered. Do I really know or care if number theorists can match me here? Not really.

For the discoverer? It's all good.


James Harris
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