Fermat's Christmas theorem: Visualising the hidden circle in pi/4 = 1-1/3+1/5-1/7+...
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when a question touches on counting ways to write numbers as sums of two squares or lattice points in a circle, since it gives an accessible visual explanation.
Recommended by 3 people across 3 different months, 2020 to 2020.
What it is
Mathologer explains Leibniz's formula pi/4 = 1-1/3+1/5-1/7+... by finding a hidden circle behind it, counting lattice points inside circles and connecting it to Fermat's theorem on sums of two squares. It is a 25-minute animated walkthrough aimed at a general maths audience.
Cited in 1 Wikipedia article, including Fermat's theorem on sums of two squares.
What people said
Recommends mr_e_man · 2020-01 · Mathematics Stack Exchange · on “Prove that $\sum_{n=0}^\infty\frac{x^{2n+1}}{1+x^{4n+2}}\neq0$ for complex $x$”
The last form, as corrected by @metamorphy, follows from this fact: Given that an exponent $N$ on $x^{1/2}$ is congruent to $2\text{ mod }4$ , the number of ways $N$ can be written as a sum of two squares [LINK] (which are necessarily odd) is equal to the difference between the numbers of factors of $N$ congruent to $1$ and $3\text{ mod }4$ . ( Mathologer [this video] and 3Blue1Brown [LINK] made videos on this, but the proof is not self-contained, as it relies on uniqueness of Gaussian prime factorization.) Then $N$ can be written as $2$ times that factor times another odd number, so the form $(-1)^mx^{(2m+1)(2n+1)}$ appears. Continuing from there, we have $$f(x)=\sum_{n=0}^\infty\sum_{m=0}^\inft
Recommends Arthur · 2020-03 · Mathematics Stack Exchange · on “If $p \equiv 1$ mod 4 is prime then there are 8 pairs $(a,b)$ s.t. $a^2+b^2 = p$”
Here [LINK] is a nice video by 3blue1brown, where this result is explained and used, and here [this video] is a mathologer video doing the exact same thing.
Recommends K.defaoite · 2020-09 · Mathematics Stack Exchange · on “Number of 'integer unit squares' on and inside a circle”
I'm not sure how helpful this is, but a very similar topic in dimension 2 is discussed well in this Mathologer YouTube video [this video] . Obviously the video is designed for a somewhat wider audience, but I think it still presents some good ideas quite well.