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 + 1128² + 1880² + 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 +...+s_c² + 1

x² + (s₁² +...+s_c²)y² = mⁿ

The BQD Iterator is:

Given nonzero integers u and v with

u² + (s₁² +...+ s_c²)v² = F

then it must also be true that

(u - (s₁² +..+s_c²)v)² + (s₁² +...+s_c²)(u + v)² = (s₁² +...+ s_c² + 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)² + 25*(-1504)² + 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)² + 25*(376)² + 49*(376)² + 121*(376)² + 169*(376)² = 189⁴

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

34967² + 752² + 1128² + 1880² + 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