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@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

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

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

@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 ok but we can't both do it

@andrewt It's yours.

@andrewt Have you submitted your talk for MathsJam yet?

Andrew 🔢@andrewt@mathstodon.xyz@ColinTheMathmo Oh, that's lovely.