Let's prove a simple "prime number theorem".

[@16sumo41]

would love it if mathmajor made like a course on analytic number theory!

[@soyoltoi]

Many of the techniques have overlap with analysis of algorithms (analytic combinatorics)

[@Alan-zf2tt]

Michael brings an enthusiasm for math that is rare to find - and it seems infection over the internet. And that can not be too bad at all, at all 🙂

[@goodplacetostop2973]

6:12 6:41 🤧 17:15 Good Place To Stop

[@Maths_3.1415]

6:12 6:41 Good place to sneeze

[@matematicacommarcospaulo]

Good place to change the chalk for the whiteboar markers.... Nobody knows how much I had theses issues during the recording for my channel 😂

[@matematicacommarcospaulo]

8:37 I've been missing your magical knocks on the board

[@maraceoofceos1243]

another really great vid, Michael. love your content

[@Uranyus36]

for the last step in the strong induction proof, if we have a strict inequality sign during the deduction, isn't it that we must use the strict inequality in the result? that means in the end P_{k + 2} < F_{k + 1} but not P_{k + 2}

[@TheEternalVortex42]

There is the "elementary" Selberg/Erdős proof of PNT which you could perhaps fit into a single (long) video.

[@schweinmachtbree1013]

here "elementary" refers to not using complex analytic methods - the Selberg-Erdős proof is still by no means elementary in the usual sense of the word.

[@hybmnzz2658]

​@@schweinmachtbree1013kind of annoying there is no common word used outside math for "simple techniques but long train of thought".

[@ChuffingNorah]

And that's a good place to stop sneezing!

[@ВалерийГайнанов-и5г]

I like the way you’re trying to make the content a little faster to consume, but I think text must be on the screen longer, at least it should stay until Michael finishes talking about it, otherwise (for me at least) you have to pause to read the text and than listen to the explanation

[@kkanden]

bless you

[@anggalol]

There is more simpler prime number theorem: π(n) ≥ 0 for all natural number n.

[@euleri0]

Very interesting, ahem you are okay.

[@JohnDalbec]

Any similar result where F_n are Fibonacci numbers?

[@tolkienfan1972]

You said "Li of x over phi of x", instead of "phi of a", which was on the screen

[@eauna0029]

Ah yes he definitely was reading it from the screen when it appeared floating in front of him, sad that he missed that

[@gp-ht7ug]

Please more videos on analytic number theory and the Riemann Zeta function Thanks

[@pierreabbat6157]

I thought at first you were going to talk about Fibonacci numbers.

[@Anonymous-zp4hb]

Notice that Bertrand's postulate implies Lemma#1? I get it though. You wanted the 'easiest' prime number theorem. So you aren't going to assume something like that.

[@Milan_Openfeint]

I wonder how hard is it to prove p_{n+1} < 2*p_n . From there, showing the main result should be easy.

[@eauna0029]

I mean it's well-known that there is a prime between n and 2n for every n (iirc, but i'm pretty sure) but i believe the proof is not the simplest there is

[@yenhai2457]

the fact that p(n)

[@ikarienator]

Now I'm genuinely concerned

[@seanbastian4614]

After watching the video, my only question is for what n is p_n+1==F_n?

[@angel-ig]

Never. You can check that the inductive proof works if you replace >= with >

[@seanbastian4614]

@@angel-ig Then why didn't he just put >? It would've made for a stronger argument.

[@robertveith6383]

@ sean bastian- The subscript is n + 1, so in writing it in this horizontal form, it goes inside grouping symbols. The question could look like the next sentence. "For what n, if any, does p_(n + 1) = F_n?"

[@seanbastian4614]

@@robertveith6383 i know the subscript is n+1

[@angel-ig]

@@seanbastian4614 Yeah, that was what I was wondering... At least he put > in the thumbnail

[@georgesanxionnat5054]

What does a proof use to go the Orem ? A lemmasine !!

[@marc-andredesrosiers523]

please use more constructive proof where they exist proof by contradiction are 'always' suspicious, because it needs to assume that the rest of the assumptions are true The consistency of mathematics as large bodies of propositions is ... an interesting problem.

[@TheEternalVortex42]

Michael actually used a "proof by negation" in this video which is perfectly valid in constructive math. (Non-constructivists call both "proof by negation" and "proof by contradiction" the same thing but they are actually distinct). Basically to prove ¬P you show P → False (this is often the most straightforward way to prove ¬P). In fact the whole proof in the video is extremely constructive. It actually gives a method to construct the n+1 prime numbers less than π(F_n).

[@pyropulseIXXI]

I take it you are an intuitionist? Which is quite odd, as intuition tells me that a proof by contradiction is just as valid as a fully constructed proof. Also, this was a constructed proof A proof by contradiction is: To prove P, show that ~P is false A proof by negation: To prove ~P, show that P is false In classical logic, P and ~~P are the same, so proof by contradiction and proof by negation are the same. But in logics where P and ~~P cannot be freely interchanged, the two are not the same

[@pyropulseIXXI]

@@TheEternalVortex42 Just like the proof that shows the sqrt(2) is irrational is a proof by negation and is perfectly valid in constructivist mathematics