Why was this visual proof missed for 400 years? (Fermat's two square theorem)
Mathologer・2 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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