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Colin the Mathmo

@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 X2+Y2 )/N

=

(total number of (x,y) for which 0<x2+y2<=N )/N

=

(total number of (x,y) inside the circle
X2+Y2=N, minus 1)/N

=

(area inside the circle X2+Y2=N + O(N) ) /N

=

(area inside the circle X2+Y2=N2, +O(N)) /N

= πN2+O(N))/N

= π+O(1/N)

Jul 19, 2018, 10:48 AM·Public·Web
4boosts·4favorites
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Colin the Mathmo

@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.

Public
Andrew

@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.

Public
Colin the Mathmo

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

Public
Colin the Mathmo

@andrewt Have you submitted your talk for MathsJam yet?

Public
Andrew

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

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