The Circle Hidden in the Leibniz-Madhava Formula

TLDRInvestigate the Leibniz-Madhava formula and discover the hidden circle within it. Understand how the formula relates to the area of a circle and learn about Fermat's Christmas theorem. Explore the 4(good - bad) theorem and its connection to Jacobi's two-square theorem. Dive into the beautiful ideas behind these theorems, although a full proof will not be included in this video.

Key insights

The Leibniz-Madhava formula, which relates to pi, is a circle thing despite being made up of odd numbers

The Leibniz-Madhava formula follows from the area formula of a circle and a result known as Fermat's Christmas theorem

The 4(good - bad) theorem, also known as Jacobi's two-square theorem, plays a role in understanding Fermat's Christmas theorem

The number of ways to write an integer as a sum of two squares can be determined by classifying the integer as good or bad based on its factors

The bad odd numbers cannot be written as a sum of two integer squares, while the good odd numbers can

Q&A

What is the Leibniz-Madhava formula?

The Leibniz-Madhava formula states that PI/4 is equal to 1 - 1/3 + 1/5 - 1/7 + ...

What is Fermat's Christmas theorem?

Fermat's Christmas theorem states that prime numbers that leave a remainder of 1 when divided by 4 can be written as a sum of two integer squares, while prime numbers that leave a remainder of 3 cannot

What is the 4(good - bad) theorem?

The 4(good - bad) theorem, also known as Jacobi's two-square theorem, provides a way to determine the number of ways to write an integer as a sum of two squares based on its factors

How are the Leibniz-Madhava formula and the 4(good - bad) theorem related?

The Leibniz-Madhava formula can be understood in terms of the 4(good - bad) theorem, with the good and bad odd numbers playing a role in the calculation of the number of ways to write an integer as a sum of two squares

Can you provide a proof of these theorems in the video?

The video will explain the ideas and concepts behind the theorems but will not include a full proof. The proofs of these theorems are complex and require further mathematical background.

Timestamped Summary

00:05Introduction to the video and the investigation of the Leibniz-Madhava formula

02:46Explanation of the connection between the Leibniz-Madhava formula and the area formula of a circle

09:09Introduction to Fermat's Christmas theorem and the classification of odd numbers as good or bad

10:19Explanation of the 4(good - bad) theorem and its connection to Jacobi's two-square theorem

19:48Proof that bad odd numbers cannot be written as a sum of two squares

22:18Closing remarks on the beautiful ideas behind these theorems and the complexity of their proofs