I think "arbitrary largeness" works better than saying "infinite gap". Infinite gap seems to suggest that the next prime is infinitely far away (which never arrives)

That large gap using factorials was really cool and surprisingly simple

I would love to see a video covering math in the more "geometric" way that pre-algebraic mathematicians understood it.

The (p + 1)(p - 1) reveal just is incredibly well executed. You had me physically sitting on the edge of my chair, to serve me one of the most satisfying possible eureka moments. Brilliant!

Your enthusiasm for Maths just rocks. I will be endeavouring to watch all your videos ....thanks so much

You remind me so much of one of my astronomy professors back in my college days when I was studying physics a few years ago. It was my first semester, my entire school life up to that point had been in the rural South, and he was from Britain, but that guy came to lecture with the same enthusiasm you show every time I saw him. It was awesome, and the first time I really saw someone teaching with that level of enthusiasm.

by my personal gut feeling, there probably SHOULD be infinite twin primes. but my gut feelings are hardly a proof, but I'm just pointing out how it would be more surprising if there were finite pairs of twin primes

Assuming the Elliott-Halberstam conjecture, there exist infinitely many primes with a gap of at least 6. This was shown in the same Polymath project. Without the conjecture (which isn’t proven yet), the gap is at least 246 as shown here. Terence Tao has been instrumental in pushing this forward.

I like how the 24 rule only work for every prime >= 5 but when you try looking at 2 and 3, you get 3 and 8, which multiply to make 24

That was one of the best infinite primes proofs explanations I've seen... love this guy!!.. instant subscribe. Brady Haran - get this guy on your show before he overtakes! : )

I just found this guy and already love him! He is always smilling and is happy to teach the stuff he likes!!! Keep it up bro :D

Love the enthusiasm, keep it up!

That is certainly a more concise way of proving the p^2-1 is a multiple of 24 than what I learned. The way I learned it you considered the primes modulo 6. So 0,2,4 mod 6 are obviously composite (ignoring 2) so there are no primes other than 2 that are 0,2,4 mod 6. Similarly 3 mod 6 has to be a multiple of 3 and thus only 3 is a prime in the case of 3 mod 6. Thus you only needed to consider 1 mod 6 and 5 mod 6 or in other word numbers in the form of 6n+1 and 6n+5 where n is a natural number. So for the 1 mod 6 case p^2-1 = (6n+1)(6n+1)-1 = 36n^2+12n+1-1 = 36n^2+12n which when you take mod 24 you get 12n^2+12n which when n is even you get that 12n^2 = 0 and 12n = 0 mod 12 and when n is odd you get that 12n^2 = 12 and 12n = 12 mod 24 which both result in 12n^2+12n = 0 mod 24 meaning that for all numbers that are 1 mod 6 n^2-1 is a multiple of 24. Similarly for the 5 mod 6 case p^2-1 expands to (6n+5)(6n+5)-1 = 36n^2+60n+25-1 = 36n^2+60n+24 which when taken mod 24 results in 12n^2+12n which we have proven above is a multiple of 24 for all integers. Thus we have proven that for all numbers n where n mod 6 is 1 or 5 then n^2-1 is a multiple of 24 and thus all primes p that are not 2 or 3 satisfy the condition that p^2-1 is a multiple of 24.

I love Euclid’s proof that there are infinitely many primes. And when I realized 2 wasn’t (always) prime (it’s not a Gaussian prime) my whole conception of math started to pivot.

You should do something on the axiom of choice. It leads to some really bizarre theorems and proofs.

Top-tier channel. Glad to be here before the algorithm catches on and boosts you to a mill.

I like the content. Don’t understand the clocks but it’s a vibe. Nice videos. Subbed!

I really appreciate the way you portray the information; it is incredibly engaging.

10:26 gave me a quick glimpse into the reality that primes are infinite in a way I never saw before.

You're fascinated by the same things I'm fascinated by! Keep this up, I've been eating these classes up for the past couple months.

Did we ever move on from math to “other units”? I love the math topics you present, btw, and I’m sure I’d love other subjects, too. But I’m definitely here for the math!

0:41 Considering the opening line of 1984, perhaps it's good that 13 isn't on clocks.

Thank you for sharing your passion!

I love this dude's energy.

I love how enthusiastic you are😊

i literally love this guy

Fun fact: The arbitrary largeness of the gap is equal to n-2 (where n is the factorial number)

Dude you are killing it! Nice Work your videos are super powerful. Excited to watch the channel grow, and see other people stoked on patterns in math.....

I just started watching your channel. Every episode has left me saying what the fuck. I subbed.

I love your videos, they're really fun and talk about stuff I NEVER hear about in math. I'm in Linear Algebra right now and most of this stuff has never been mentioned. Did you have a formal math education or did you just find this stuff yourself?

the prime fact that still melts my brain and i don't want to believe is that for any finite integer, you can find a gap between two primes of that size. that means you can pick ANY arbitrarily large number, even Graham's number, and then if you start listing all positive integers, eventually you will find a prime, and then Graham's number of non-primes, and then another prime. Basically the only way this makes any amount of sense is because there are infinitely many integers and infinitely many primes in that realm. If you have infinite time to count, you can find a chain of non-primes of any finite length because the length is finite. There is no "last prime" though, so the chain has to be finite. but it can be Graham's number!!! that shouldn't be possible! The location of that chain may be as far out as inverse log* of graham's number, but its there. And its practically impossible to find! there is no conceivable way that any sentient species could count fast enough to find it. you would essentially have to be God to know where that chain is in the integers. Or rather, if you found the chain, it could only be the result of god-like powers.

dude this channel is going places.

Random ponder: is it a problem for mathematicians that, when you talk about infinite things, it's sometimes ambiguous if you mean "THE infinity" vs "something else that also has the property of being infinite"?

This episode really put into context for me what infinite numbers really means, cause its easy to just say numbers are infinite because n+1 always works but that doesn't really make you realize what the scope of numbers and math really is, but realizing that not only are primes infinite but that their gap can be arbitrarily large really put it in perspective, the fact that one infinite set of numbers can be spaced so far apart within another infinite set of numbers just kind of boggles my mind.

(.75*(x^2))+(1.5*x)+23 = mostly Eisenstein primes or semi-primes, when x is an even number.

just watched two of your vids and given a sub i think youre gonna blow up mate fantastic content

Humor me... can you show this sequence evaluation with an 11 integer/12 integer sequence!? Prime interests lol

Lovely presentation!

1:00 I can tell grammar isn't his *prime* subject (pun intended)

Once I imagined something somewhat like the primorial, and now I know it exist. Also you can think of the biggest primorial as the most antiprime (or prime composed) number, I guess.

Doesn't infinite prime proof, one at 5:00, also prove that there are infinitely many twin primes? since P#-1 and P#+1 are both primes and one is 2 more than the other making them twin primes?

Can you teach me, how to determine whether the number is a prime or not by using the fastest method??

I think that if you can prove whether or not there is a lower bounding number for the distance between infinite amounts of prime pairs, then you can prove whether that bound is 2. If the proven bound keeps shrinking, then there really isn't any reason to believe that the ultimate bounding number isn't 2.

Every number comes from a primal backbone. Except for twin numbers that are close to number squares, so we have the same number being the divisor of two consecutive numbers And in numbers out of order in general. And at the same even and odd numbers without corresponding prime divisors in the same number.

I have to point out that the thumbnail for this video looks like a rap music video

Are prime numbers the only sequence where the nth term is unpredictable?

I think I have a proof for twin primes. Take P# where P is a large prime number. P# + 1 must be prime, for reasons explained in the video (around the 5:00 mark, I think.) P# - 1 must also be prime for the same reasons. (P# + 1) - (P# - 1) = 2. Twin primes are infinite.

This channel is so criminally underrated

With the mod vid explaining base 10 and the speciality of whatever the mod is minus 1, so 0 for mod 10... This prime multiple of 24 + 1 bit has me questioning what happens when you use a mods with prime numbers as the base...?! Anyway.. Jus tfound you today. and I stank at maths and hated it in school. But do find it very interesting. Been watching numberphile for years and mostly understand very little of it.. But love the way you break things down., So will def be watching more.

"We could make every number" except 1 >: )

I love your videos

Can Prime be The number itself? Like nothing bigger than Prime itself, and all other things are expressions of it.

Hi

The Fundamental Theorem of Arithmetic should have been called the Fundamental Arithmetic Theorem. For the acronym.

Fun.

Nice Tiktoks too wow thanks♾♾♾♾♾☮️💟😘🥰😍😻🤩🤯🗽🌈

PLEASE TALK ABOUT THE REIMOND HYPOTHESES SOMETIME

10x9x8=6×5×4×3×2×1

You are insane, and I'm okay with that

Wow, look at that lab-coat. So white, fresh and unscorched. 💀

I suspect the twin prime conjecture may have some correlation to the arbitrarily large gaps by n! due to the fact that, while you only showed the numbers that get blocked by ADDING, it actually blocks them by subtracting too. So, n! blocks primes from existing from (n! - n) all the way to (n! + n), with the only exceptions being n! +/- 1, which, if both end up being prime, would automatically be a twin prime. Of course, it's clearly not that simple to PROVE or that would have already been proven. But I suspect if there is an "easy" solution to it, it'll be found in some sort of connection to that fact. Also, just for fun, if you could push n to infinity (say by using limits in Calculus) then n! would have every single possible number as a factor. This means that n + 1 would necessarily have no factors at all, except for 1 itself. And the only number that has just 1 as a factor is 1 itself. Thus, implying that n! = 0 when n hits the limit at infinity.

Does not the Fundamental Theorem of Arithmetic (FTA) _assume_ numbers/integers (and hence prime numbers) are infinite? In which case, using FTA to prove there are an infinite number of primes is a circular argument.

0 isn't prime because it's even. Sorted.

As an example of the infinite primes proof: 4! = 24 24 + 1 = 25 = 5*5 5 is not a factor of 4! 24 - 1 = 23 which is prime. With primordial: 5# = 2 * 3 * 5 = 30 30 + 1 = 31 > 5 and is prime. QED

4% of numbers, at most, are prime.

Um. Pz# has no factors, since Pz is prime.

Here is an ethical question for you. Would you hand a 3 year old child a loaded shotgun? If we develop the math correctly (people have been getting close) and accept that math's paradigm shift (hell, the US cannot even get the metric system imposed... but hey, the world thinks that E-MCC was discovered by Einstein and Trump won the 2020 election.). then we will be able to make things that make the atomic bomb look like a birthday candle. Do we really want the people we elect to have such things?

Not unhinged. Unwatchable.