Infinite Anti-Primes (extra footage) - Numberphile

Featuring Dr James Grime.

NUMBERPHILE

Videos by Brady Haran

[brianpso]

This ending was so perfect. Dr Grime is just amazing to watch.

[tone]

Wouldn't it be easy to prove by contradiction that there were infinitely many? If there's not, then there must be a highest one, and if there's a highest one that implies that there's a maximum number of factors that a number can possibly have, which seems inherently nonsensical.

[YuTe]

I have this unfounded feeling that Ramanujan left it out on purpose. :)

[otakuribo]

Matt: "Stop trying to make Parker Square a thing!"
Brady: *make shirt of it
*Parker Square becomes a thing

James: "Stop trying to make Anti-prime a thing!"
Brady: *puts anti-prime in all titles and thumbnails in bold font
*???

[pegy]

293, 318, 625, 600 must have felt betrayed by its friend, Ramanujan, to be left off of his list.

[jeffirwin]

Ramanujan made a serious Parker Square with 293, 318, 625, 600.

[Miker]

Mind blown at the ending. Gr8 finish. Strong.

[saulivor]

oh Ramanjan, how could you miss 293 Billion 318 Million 625 Thousand 6 hundred

[wafelsen]

It still seems like they would be easy to build from this model and seeing where to add the next prime factor adds factors faster than it increases the number (a bit like the Matt Parker problem from 2020 about a marching band with 64 arrangements)

[chuvzzz]

That seems like too much of a coincidence. Maybe he left it out as a joke, or a little treat, for whomever were to check his work

[Pfisiar]

Proving that there are infinitely many primes and proving that all composite numbers are products of primes would seem to, logically, prove that there are infinitely many highly composite numbers.

[apeman]

If anybody is curious, proving that there are infinitely many highly composite numbers is relatively simple - about at the same difficulty as proving there are infinitely many primes. Give it a shot!

[JM-usfr]

Ok, but I still want to know why the last prime has a power of 1

[Bitflip]

Multiplying a highly composite number n by 2 doesn't necessarily create another highly composite number (as you show). But even if it doesn't, then there must be another number between n and 2*n that is highly composite.

This is because multiplying any number by 2 creates a number with at least 1 more factor. If this new number isn't highly composite, then there must be a number smaller than 2*n that has at least as many factors as 2*n. In that case, it will have more factors than n so it must be greater than n because n is highly composite.

So the hypothesis involving multiplying by 2 isn't true. But it does create an upper bound of sorts.

[SmileyMPV]

I love these numbers already :) Been thinking about them for a day now and I have come up with some cool proofs, including:

1) There are infinitely many highly composite numbers.
2) For every natural number, there are only finitely many highly composite numbers that are not divisible by that number.
3) There are only finitely many highly composite numbers where the last power in their prime factors is greater than one.

Ideas 1) and 3) were taken from the Numberphile videos and 2) is used in proving 3).

[jwmmath]

...'old' favorite anti-Prime # was 55440. Favorite today is 360360. Easily shown as 360 x 1001, and LCM of 1 thru 15.

[Austinwolf]

We have gotten a biggest prime video but what's the biggest highly composite number that's been found

[Kram]

I noticed that the first couple highly divisible numbers all had the previous largest highly divisible number as their number of divisors. Of course this happy circumstance breaks down eventually. Still, it's quite curious. So I wondered, is there anything to this? Do highly divisible numbers always have a highly divisible number of factors (not necessarily the previous one) or do they at least have that more often than non-highly-divisible numbers?

[nicks]

So...

4 is a highly composite number.

The smallest number with 4 divisors is 6, which is a highly composite number.

The smallest number with 6 divisors is 12, which is a highly composite number.

The smallest number with 12 divisors is 60, which is a highly composite number.

The smallest number with 60 divisors is 5040 which is a highly composite number.

The smallest number with 5040 divisors is 293318625600 which is a highly composite number.

I feel this is somehow deep but I’m not sure why.

[glathir]

Wow, how could someone miss one as obvious as 293, 318, 625, 600? You can just look at that and tell it's highly composite! :-P Seriously though, does anyone know what Ramanujan's method was, or did he just have a lot of spare time, when finding these numbers?