Why was this visual proof missed for 400 years? (Fermat's two square theorem)
Mathologer・24 minutes read
Fermat's square theorem explains how primes of the form 4k+1 can be written as the sum of two integer squares, with a recent visual proof using windmills to demonstrate the odd number of solutions for these primes. The proof involves a unique windmill configuration for 4k+1 primes, showcasing the elegance and simplicity of Fermat's theorem.
Insights
- Fermat's Christmas theorem, also known as Fermat's two square theorem, focuses on expressing primes as the sum of two squares of positive integers, with successful primes being of the form 4k+1 and unsuccessful ones of the form 4k+3.
- A recent visual proof of Fermat's theorem, shared by Mathologer, showcases the odd number of solutions for a 4k+1 prime through a unique windmill interpretation, highlighting the concept's elegance and the role of windmill configurations in proving the theorem's uniqueness.
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Recent questions
What is Fermat's Christmas theorem?
Fermat's Christmas theorem, also known as Fermat's two square theorem, is a crucial result in mathematics that plays a significant role in deriving the circle-free pi formula. It involves writing primes as the sum of two squares of positive integers.
Why is Fermat's theorem considered beautiful?
Mathematicians voted Fermat's theorem as the tenth most beautiful theorem ever in a scientific poll due to its elegant nature and the intriguing way it allows primes to be expressed as the sum of two squares of positive integers.
How are primes represented in Fermat's theorem?
Primes in Fermat's theorem are represented as the sum of two squares of positive integers, with a clear pattern emerging where successful primes are one up from a multiple of 4, while unsuccessful primes are of the form 4k+3.
What is the key observation in the proof of Fermat's theorem?
The key observation in the proof of Fermat's theorem is that for any 4k+1 prime, there is always an odd number of solutions to the equation, leading to the successful representation of primes as the sum of two integer squares.
How does the windmill interpretation aid in understanding Fermat's theorem?
The windmill interpretation provides a visual understanding of Fermat's theorem, showcasing how a 4k+1 prime can be written as a sum of two positive integer squares in exactly one way. The unique windmill configurations play a crucial role in proving the theorem's concept of expressing primes as the sum of two squares.
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Summary
00:00
Fermat's Christmas Theorem: A Visual Proof
- Fermat's Christmas theorem, also known as Fermat's two square theorem, is a significant stepping stone in deriving the circle-free pi formula.
- Mathematicians voted Fermat's theorem as the tenth most beautiful theorem ever in a scientific poll.
- Fermat's theorem is about writing primes as the sum of two squares of positive integers.
- Primes of the form 4k+1 can be written as the sum of two integer squares, while those of the form 4k+3 cannot.
- A simple pattern emerges where successful primes are one up from a multiple of 4, while unsuccessful primes are of the form 4k+3.
- Euler's proof of Fermat's theorem involved complex mathematical concepts and was challenging to understand.
- A recent super simple visual proof of Fermat's theorem was discovered and shared by Mathologer.
- The proof demonstrates that primes of the form 4k+1 can be written as the sum of two integer squares.
- The proof involves considering a related but more general problem to gain insight into the original problem.
- The key observation in the proof is that for any 4k+1 prime, there is always an odd number of solutions to the equation, leading to the proof of Fermat's theorem.
15:08
Odd Solutions in 4k+1 Primes Demonstrated
- If the total number of solutions is odd, there must be an additional solution of the special y equals z type.
- Fermat's square theorem can be proven by showing that for any 4k+1 prime, the general equation has an odd number of solutions.
- A visual representation using windmills helps demonstrate the odd number of solutions for a 4k+1 prime.
- Windmills represent solutions to the equation, with each windmill having a total area corresponding to the prime number.
- Swapping values in the windmill configurations reveals additional solutions, always resulting in an odd number of solutions.
- The footprints of the windmills come in congruent pairs, except for the straight cross windmill, which occurs only once.
- For a straight cross windmill, the side length of the square is equal to the side length of one of the rectangles, leading to a unique solution.
- The windmill pairing method proves that there is an odd number of solutions for any 4k+1 prime.
- The windmill interpretation of Zagier's proof provides a visual understanding of the mathematical concept.
- A 4k+1 prime can be written as a sum of two positive integer squares in exactly one way, with the windmill configuration playing a significant role in the proof.
31:07
Unique proof for 4k+1 prime sum.
- There is only one way to write a 4k+1 prime as a sum of two integer squares, with the proof involving a uniqueness argument and the utilization of well-known mathematical identities.
- A challenge is presented to show that a nonzero integer can be expressed as the difference of two integer squares if the integer is odd, and this representation is essentially unique for prime odd integers. The text also expresses gratitude towards supporters of Mathologer.
