The Prime Number theorem has several important consequences, including the infinitude of primes and the fact that the density of the primes is 0, which is to say the limit of the prime counting function at x divided by x approaches 0 as x approaches infinity. Today we give a fairly elementary proof of this consequence of a very non-elementary theorem. The proof we show that the primes get arbitrarily sparse does not rely on the prime number theorem. #maths #mathematics
Proof due to Idris D. Mercer: https://www.idmercer.com/primes-density.pdf
Join Wrath of Math to get exclusive videos, lecture notes, and more:
https://www.youtube.com/channel/UCyEKvaxi8mt9FMc62MHcliw/join
More math chats: https://www.youtube.com/playlist?list=PLztBpqftvzxXQDmPmSOwXSU9vOHgty1RO
Textbooks I Like!
Graph Theory: https://amzn.to/3JHQtZj
Real Analysis: https://amzn.to/3CMdgjI
Abstract Algebra: https://amzn.to/3IjoZaO
Linear Algebra: https://amzn.to/43xAWEz
Calculus: https://amzn.to/3PieD1M
Proofs and Set Theory: https://amzn.to/367VBXP (available for free online)
Statistics: https://amzn.to/4d5HbUQ
Discrete Math: https://amzn.to/3qfhoUn
Number Theory: https://amzn.to/3JqpOQd
Algebra 1: https://amzn.to/3Z4SPvu
Algebra 2: https://amzn.to/3LWC7XD
Precalculus: https://amzn.to/4fO4On9
★DONATE★
◆ Support Wrath of Math on Patreon: https://www.patreon.com/join/wrathofmathlessons
◆ Donate on PayPal: https://www.paypal.me/wrathofmath
Outro music available for channel members.
Follow Wrath of Math on...
● Instagram: https://www.instagram.com/wrathofmathedu
● TikTok: https://www.tiktok.com/@wrathofmathedu
● X: https://x.com/wrathofmathedu
0:00 Intro
0:42 Prime Number Theorem and Consequences
3:09 The 4 Keys
3:56 1) pi(x)
4:54 2) 999+1
7:50 3) Sets, Subsets, and Binomial Coefficients
10:57 4) Logx
12:42 Proof Part 1
18:22 Proof Part 2
22:39 Proof Part 3
25:02 Proof Part 4
28:32 Proof Part 5
32:47 Proof Part 6
37:49 Finish