Sums of squares for same value

Handling sums of squares can be mostly automated using some of my mathematical research. For instance consider:

5² + 20² = 17*5²

8² + 19² = 17*5²

13² + 16² = 17*5²

These were found using what I decided to call a Binary Quadratic Diophantine iterator, or BQD Iterator for short, which lets you find in general solutions for x and y with:

x² + (m-1)y² = F*mⁿ

When F = x₀² + (m-1)y₀², my research proves there are non-zero integer solutions for x and y, where n is a count of iterations.

To get my examples used x₀ = 1, and y₀ = 2, and picked m = 5 so that I had 4 which I could pull into the square, to get a sum of two squares.

At each iteration you get a split point, where you can go positive or negative, which means as you iterate you may generate extra solutions for the same sum.

For my example I had a duplicate in the second iteration which is why there are only 3 distinct solutions instead of 4.

And the 5 being squared in this example is not an accident, as that's the second iteration. The first iteration has 17*5.

So yeah I could have kept going, and would have had a maximum of 6 cases of two squares summing to equal 17*5³, but less if there were duplicates.

I have an earlier post which shows how the solutions were found.

And I have additional research using the BQD Iterator showing how you get a desired number of sums of squares.  Which could mean for math hobbyists, maybe could be useful in generating magic squares of squares?


James Harris