Do 5

Even though I know about number theory, and know about perfect, abundant, deficient, amicable, sociable, I had never heard of aspiring numbers before. Now because we have names for all of these categories, we seem to need one more. Doing the aliquot process once, divides all numbers into three categories: deficient, abundant, and perfect. But doing an aliquot sequence, we get (potentially) seven categories, but three of them don't seem to have names: Perfect - stay the same forever. Aspiring - eventually get to a perfect number. Amicable - bounce back and forth between two values. Sociable - cycle through a loop of more than two numbers. ?1? - the ones that never get to a loop or perfect number - there might not be any in this category. ?2? - numbers that eventually get to a loop. You might say they "aspire to be amicable or sociable, rather than aspiring to be perfect". ?3? - the numbers that get to 1 eventually. Note that both abundant and deficient numbers can fall into this category. I guess those ?1? numbers, if they are found to exist, can be named after whoever finally proves their existence. The ?2? numbers could be called "shy" numbers - they're trying to get into the amicable/sociable group. I suppose this category could be split into two. And the ?3? category in which the majority of numbers fall, should have some name, too. At first, I was thinking to propose calling them "mortal" numbers, because through the aliquot sequence, they eventually "die". But that seems too dark of a name.

The next puzzle for you to solve: The 300 Coins Problem. 300 coins are placed randomly on a table. A 300 letters long message (Signal) is written, one letter per coin, that would lead to a hidden treasure. Then the coins are flipped over and a randomly generated Noise 300 letters long is written on the other side of coins. The coins then get put in a bag and scrambled. Finally, the coins are put back on the table. Your task is to flip and move the coins around until the original message is recreated. Can you do it?

I checked wikipedia on sociable numbers for my own curiosity, and if it is accurate then: The only known loop lengths are 1 (perfect), 2 (amicable), 4, 5, 6, 8, 9 and 28. (and 5, 9 and 28 only have 1 known sequence each) "It is conjectured that if n is congruent to 3 modulo 4 then there is no such sequence with length n." So loops with length n=4k+3: 3,7,11,15,... is probably/maybe not possible.

U still exist?

That's the plan

Absolutely beautiful and simple conjecture! And i love that that's the plan!

the numberphile playlist of all videos will then become an OEIS sequence since it will have a unique sequence of integers by age of video.

Fun fact: 9538 is the smallest number that can't be defined in 30 English words or less.

@@thewhitefalcon8539”Nine thousand five hundred thirty-eight”

something something keychain parties

I think he's ending up in the dog house for a while ;)

Time stamp?

4:58

284

Was it just a brainfart, or did you think about 296 for different reasons and got it mixed up?

I think I had 496 in my head (for perfect reasons) and it contaminated my thoughts. Mea culpa. 🫤

Epic fail )

🫂

@@NorlanderGT the answer is at 4:02

Here before this comment is popular

masterpiece

It was the "go son!!" That sent me

They need him in as a guest commentator on @jellesmarbleruns

Made my day and it's not even 8am!

Didn't James Grime mention this as a thing to do when he taught us amicable numbers like 10 years ago?

Here before this comment is popular

​@@HasekuraIsuna I wonder if that's where Ben got the idea from.

So romantic to forget your wife's number

You can buy the keyrings at Maths Gear.

in the midst of "its so over", I found there was within me, an invincible "we're so back!"

I looked specifically for this comment

Brought to you by... Jelle's Marble Runs!

Reminds me of Tetris gameplay shooting for some crazy world record breakthrough.

WHEEEEEEE

I was about to say!… the path for 138 looks like a stock price.

The universe doesn't care about intuition 😂

@@daniel_77.your comment makes no sense

@@vikashchandra9917 Sorry. I meant that the things we see and do, even the seemingly simple natural numbers, still hides a lot of complex reasoning. When things may seems obvious and intuitive, In reality it doesn't work like that.

This is the video I'm going to cite for the foreseeable future when someone asks what number theory is. And I'm going to foist it on my kids tonight

Just need Tadashi Tokeida to incorporate some weird toy into it

+

One wonders if 276 and 552's aliquot trajectory would ultimately be in some way analogous to 138's, except by several orders of magnitude longer in sequence

If you look at it in terms of using 138 as a base number n, 3 of the 5 numbers are multiples of n. 2n, 4n, 7n.

@@AndyWitmyer 276 already has an index (sequence length) over 2,100 and 564 has an index near 3,500 currently. 138 ONLY took an index of 177 to resolve, thus both sequences are already at least one order of magnitude larger and show no signs of ending anytime soon.

you mathematicians don't know shi...

You noticed that, too!

It practically is, in more ways than one.

This was my thought. If you want to really chase a rabbit hole google Muratz Conjecture in relation to Collatz and you start to see real similarities. Wonder if there's something there.

You mean : the Collatz conjecture (or Hailstone or 3n+1) and variants that divide a number by 2 if it is even and else multiply it by 3 and add 1. Yes, a very similar situation that also came to my mind ( and many others I'm sure ). Great video Ben !

Lol that would be funny

I'm still recovering

​@@numberphile I think we should call them "rollercoaster numbers"

It'll be his version of the Parker square

HAHA

The Sparks Amicable

GO ON SON!!!

True, although the number 27 in the Collatz conjecture is a low number, yet blows all the way up to 9232 in a similarly shocking manner, but not quite like this! This is a more fundamental number theory, of which the Collatz conjecture is a more complex flavour.

Yeah I was shocked how low the first number is where we haven’t figured out the answer.

I think this is like Collatz, technically we don't know but (having spent a lot of time with these sequences) my suspicion is that infinity is an awfully long time for it to *not* end at some point. I think they will all end, it just takes enormous amounts of computations to check

@@TimSorbera the difference with collatz is that we know the answer for all starting numbers up to something like 2^60. It’s wild to me that we don’t know the answer for a starting number as low as 276.

@@TimSorbera Richard K Guy presented some evidence for a counter-conjecture that there are unbounded aliquot sequences.

There’s a lot of thing like that that amaze me. It’s trivial to prove that if you gather 6 people together, either you have 3 mutual acquaintances or 3 mutual strangers. 18 ensures 4 mutual acquaintances or strangers. But the minimum number to ensure 5 mutual acquaintances or 5 mutual strangers is still unknown (except that it’s between 43 and 48).

Cheers - glad you enjoyed it

It's not as slow as people say, it just isn't as blazing fast as something like C. This only really becomes apparent when you start massively scaling your computations, so you wouldn't want to run Python on a scientific supercomputer.

Fortran77 ftw

Everything is relative :)

@@hammerth1421 except it does matter a lot - when your calculations are 10x (if not 100x) slower, it means you can do 100x less on your own PC before you have to resort to clusters/supercomputers. which of course is terrible news for hobbyists, not that that _really_ matters

Threat?

@@Pathakin. Gotta love autocorrect 🙃

O.G.

what fandom

​@@emperortgp2424 Based on the account, I assume they're referring to the number 2763 being mentioned multiple times in Battle For Dream Island episodes. Hope that helps! :)

The prophecy is spoken, we must test it.

@@MathNerd1729 yes

We love the people who watch it!

Those doing research on these sequences have noticed a few patterns (the technical term is "guides") generally based on two principles: * How many powers of two the number you're looking at has (fewer means smaller numbers and more means larger numbers) * Is there any power of three in the number (if yes -> bigger numbers, if no -> smaller numbers) You generally want the terms in the sequence getting smaller because that increases your odds of it terminating by hitting a prime (or some kind of cycle of numbers).

Part of the problem is that if odd perfect numbers DO exist (many people think they don't) they're going to be very large numbers to work with - the current best estimate of the smallest one is at LEAST 2300 digits with 48 factors!

so 8 person mutual ...... leads to infinity?

Obligatory "Nice."

Write phyton code to check ❌️ Write code in geogebra ✅️

Link to the file is in the description, in case you missed it

Dare to dream.

It seems to me that what really spells an even number's doom is hitting a square or twice a square. Because then you hit odd numbers, and they are much more likely to be deficient. Hitting an odd square gets you back into even numbers again, but that's unlikely.

@@rosiefay7283 You don't even need squares; due to how the sequences are calculated, any number which has two (and only two) similar sized factors will have a dramatic drop in the next term's size, likely enough for it to hit a prime or a "local minimum" that it will start climbing up from again.