Anyone know why (or, I suppose, if) that's true? I can't find the original anywhere.


@andrewt From Gareth McCaughan.

The average number of ways to express numbers from 1 to N
as sums of two squares


(total number of ways to express numbers from 1 to N as \(X^2+Y^2\) )/N


(total number of (x,y) for which \( 0 < x^2+y^2<=N \) )/N


(total number of (x,y) inside the circle
\( X^2+Y^2=N \), minus 1)/N


(area inside the circle \( X^2+Y^2=N\) + O(N) ) /N


(area inside the circle \( X^2+Y^2=\sqrt{N}^2 \), +O(N)) /N

= \( \pi\sqrt{N}^2+O(N))/N \)

= \( \pi+O(1/N) \)

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@andrewt It is - I agree entirely. One could probably make some nice visualisations of that happening, and there are less formal (but effectively identical) arguments.

But yes, lovely.

@ColinTheMathmo I had to draw some stuff to properly get my head around it but yes, with the visuals and some handwaving it's very intuitive.

Perfect MathsJam talk material.

@andrewt Indeed - a lovely opening hook, and once seen, a clear intuition as to why it's true.

@ColinTheMathmo oh, I have not, I quite forgot. That can go on tonight's list, right after "clean kitchen"

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