Prime Number Theorem & Natural Logarithm

The Prime Number Theorem is explained. The natural logarithm is introduced using the logarithmic spiral to give you a better feeling for this type of growth. These come together to give us an intuitive feeling AND formula for how many primes exist as we look towards infinity.The Density of Primes

[brendanhuynh]

Wow. Wow. Wow. Wow. At first I thought I was watching a video made and released in 2022. Then I realized that this video is 9 years almost 10 years old. I'm seriously impressed with the quality of this video. Especially for it's time.

[KhaanTravels]

This video BLEW my mind. I am in my third year of college studying computer science. This video finally helped me connect calculus with computer science. I will defiantly come back to this video for my discrete  mathematics class.

[late_arvie]

Another stunning video. Everything was explained very well. Even for me - amateur math enthusiast.

[ozzyfromspace]

Didn’t mean to watch this, but thank you for saving a few brain cells. Humanity owes you one bud, great video!

[KhanAcademyLabs]

I used Apple Motion for most of these graphics. Everything is based on actual data. I took a Ulam spiral plot of several million primes and used a virtual 3D camera to fly around the image. The graphs were all made using Grapher which is a free os-x program

[interguy]

Wow! Even if it’s mostly conjecture, it’s incredible to see how close the correlation is!

[silencedidgood]

Brit...you are rapidly showing the brilliance of John, Vi, the Bens....etc and the world is better for it!

[episer]

Thank you for your great work! Really enjoyed it

[nuevemillas]

I suppose the blue grid is representative and not exact right? Ive seen other prime grids and the decay beyond the initial dense cluster is more significant. In this representation you see the surroundings like if primes werent decaying at all ...

[paulkestyn]

An electron traveling around a surface at a reguar growinging circumference at a regulary growing speed. The begining point relative to observer determines the Prime Spiral.

[tangoolo]

This should be The Future Of Education...mannnn SIMPLY AWESOME!!! No need of stupid fiction as mode of entertainment b'coz...REALITY IS STRANGER THAN FICTION!!!! And more captivating too.

[mattoniogaming]

Well this video just helped me majorly with my scence fair project. :) Thanks.

[pauluk]

6:00 I think the claimed accuracy is off by a factor of 100. You are taking the difference between the count of primes and the estimate and then dividing that number by 100 tn. But the true relative error of the estimate is (count - estimate)/count - or about 1/32.

[nlo]

Interesting. I notice a regular (although square) pattern of 'radial' lines coming out from the centre of the no.1 point of the square. Bearing in mind the regularity of the lines, could the numbers be spaced around a different polygon to produce a straight line? I would imagine it would look something like a Mandelbrot sequence. I'm not sharp-witted enough to do this myself. Any ideas?

[mikecmtong]

lon x? Who calls natural log "lon x?" lol

[amandeepsingh]

+Art of the Problem, Thank you so much for such an amazing explanatory videos.

[Robert_McGarry_Poems]

Yo, you have been at this fire a long time. Nice...

[jonasiye]

hi, congratulations for all your good vids. I wonder if you would mind telling me what software you use for the editing of the movie. Thanx

[FPrimeHD]

Holy shit, you are amazing. Great video!

[malkdk]

How on earth did you create those graphics? That's just awesome! Is the animation based on actual data? I mean, whenever the angle is changed, do blue dots still accurately represent prime numbers?

It would be so awesome to have a program running on my computer that could do that; map all primes as blue dots and zoom out and in on different parts and change the angle.

How do you do it?