## Interesting sum of squares problem

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• ... Hash: SHA1 Recently I ve been trying to work out matching pairs of i^n + j^n = k^n + l^n, excluding trivial solutions of course. By the way, for n = 5,
Message 1 of 1 , Mar 22, 2003
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Recently I've been trying to work out matching pairs of i^n + j^n = k^n + l^n,
excluding trivial solutions of course. By the way, for n = 5, there seems to
be no matching pair with i,j,k,l <= 10000. Maybe there are no matching pairs
for n = 5 -- would anyone care to comment?

But that was not the point of my question. In the course of parallelizing the
program, I found out a very interesting fact for n = 2. Say I'm working with
numbers <= x, so max(i^2 + j^2) = x^2 + x^2 = 2x^2. For i^2 + j^2 < x^2, the
values taken by this sum of squares are evenly distributed with a frightening
accuracy. After x^2, it appears to follow an exponential distribution, but I
don't really care. A plot of these values for a bound x = 200 (thus, linear
range ends at y = 40000) is at http://distributed.net/~acidblood/grafico.zip

I've been staring in awe at this graph for a few hours today already, but I
can't explain what's going on. Any ideas?

Thanks,

Décio
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