Fermat's Christmas theorem: Visualising the hidden circle in pi/4 = 11/3+1/51/7+...  Summary and Q&A
TL;DR
The Leibniz formula for pi can be derived from the area formula of a circle and the number of lattice points within it.
Key Insights
 🦕 The Leibniz formula connects the odd numbers used in the formula to a hidden circle.
 ⭕ The approximation of pi using the number of lattice points in a circle improves as the radius of the circle increases.
 #️⃣ The "good" and "bad" odd numbers play a crucial role in calculating the number of lattice points and have a connection to Fermat's Christmas theorem.
 🦡 The 4(good  bad) formula can be used to calculate the number of lattice points by tallying the "good" and "bad" odd numbers.
 🍹 The identity that the product of two sums of two integer squares is also a sum of two integer squares is crucial in extending the results to all integers.
 🤩 Quadratic reciprocity and Fermat's theorem are key components in fully understanding the proofs of these theorems.
Transcript
(Thanks to Karl for the 2019 Easter egg idea :) Welcome to the 2019 Mathologer Christmas video. In this video we'll investigate that famous and amazing formula over there PI over 4 is equal to 1 minus 1/3 plus 1/5 minus 1/7 and so on. It's usually called a Leibniz formula after Gottfried Wilhelm Leibniz one of the genius inventors of calculus. Sadl... Read More
Questions & Answers
Q: Who first discovered the Leibniz formula for pi?
The formula was first discovered by the Indian mathematician Madhava of Sangamagrama in the 14th century.
Q: How can the number of lattice points within a circle be used to approximate its area?
Each lattice point can be considered the center of a unit square, and the total number of lattice points approximates the area of the circle.
Q: What is Fermat's Christmas theorem?
Fermat's Christmas theorem states that "good" primes can be written as sums of two integer squares, while "bad" primes cannot.
Q: How can the 4(good  bad) theorem be used to calculate the number of lattice points in a circle?
By tallying the "good" and "bad" odd numbers and using the 4(good  bad) formula, the number of lattice points can be calculated.
Summary & Key Takeaways

The Leibniz formula, which approximates pi, was first discovered by the Indian mathematician Madhava of Sangamagrama in the 14th century, over 200 years before Leibniz.

The number of lattice points within a circle can be used to approximate its area.

By applying a theorem known as Fermat's Christmas theorem, the number of lattice points in a circle can be calculated using the idea of "good" and "bad" odd numbers.