How Goldbach and Euclid Proved There Are Infinitely Many Primes?

Take the numbers 2, 3, 5, 7, 11 and so on, they are known as prime numbers in a field of mathematics known as number theory. They only have divisors 1 and the number itself. Mathematicians from the ancient greeks have began investigating the mysteries hidden behind this sequence of numbers and one of the most fundamental properties of this sequence is that it grows to infinity, in other words, there are infinite number of prime numbers. In this video, I would like to present a classic proof of this fact by Euclid in one of his famous book, Elements (IX,20) and the another proof is by Christian Goldbach in a letter to Leonhard Euler in 1730, yes, he is the mathematician who have proposed one of the unsolved conjecture in modern number theory, the Goldbach Conjecture.
🔥Subscribe to 1Psi3Colour:
czcams.com/users/1Psi3Colour...
Video Chapters:
0:00 What are prime numbers? (and history)
0:55 Euclid's Proof
2:03 Goldbach's Proof
4:50 Outro + Subscribe!
💥Check out these videos:
The integral everyone should know: • The MOST Influential I...
A deceivingly easy Putnam problem!: • Is This THE EASIEST Pr...
👉Check out these playlists:
Math Olympiad: • Math Olympiad
Math puzzles: • Math puzzles
College Math: • College Math
This video is created using Manim:
www.manim.community
👉Suggest a problem: forms.gle/hTibrUKz7QNyqoC48

[@krasimirronkov17]

Very good video. I have one question though how can you prove the recurrence relation without using induction.

[@1psi3colourmath]

That's a great question! You can use the relation, F_n=(F_{n-1}-1)^2+1.
Then
F_n=F_{n-1}(F_{n-1}-2)+2
=F_{n-1}F_{n-2}(F_{n-2}-2)+2
=F_{n-1}F_{n-2}F_{n-3}(F_{n-3}-2)+2
=...
=F_{n-1}...F_0+2