BREAKING NEWS After 6 years, a 52nd Mersenne prime has been found, which means there is also a 52nd perfect number! I hope you like numbers because this video is extremely mathy! Thank for the patience awaiting the new video as I've been busy irl. I hope you enjoy!
I once saw someone writing a code to determine if n was a perfect number. The code computed σ(n) by checking all numbers d up to the square root of n, and adding d and n/d to a total whenever d divided n. However, when n=N² is a perfect square, the divisor d=N was not included in the sum (due to a < sign), and instead of comparing 2N² with σ(N²), the code was comparing it with σ(N²)-N. I coined the false positives that the code may yield (which are a very niche subset of the near-perfect numbers) as PSEUDOPERFECT numbers. I told the person who wrote the code that it was flawed. However, I was unable to find a counter-example. Over the years, I have checked up to n=458,335,615,276,564,171,975,521 (inclusive) without finding a single pseudoperfect number, but I can't discard that they exist. I would love to know whether they exist, because even though it's been almost 10 years, if it turns out that pseudoperfect numbers don't exist, then the code would be valid and I should apologize to that person.
I did some work on this problem. It’s well known that square numbers can’t be perfect numbers, so any square returned by the algorithm would be wrong. This is because you get (even) 2N^2 = (odd) σ(N^2). For the algorithm to return a false result, we need 2N^2 = σ(N^2) - N, which would require N to be odd to make the whole expression even. Also, we can rearrange to get 2N^2 + N = σ(N^2), or N(2N+1) = σ(N^2) for some odd N. This tells us that N | σ(N^2) and 2N+1 | σ(N^2). This last statement may lead to a contradiction, proving the algorithm always works, but my number theory is rusty so I’ll have to stop here
Some other ones I’ve came up with (others probably have found these as well): Barely Abundant: A number N whose aliquot sum equals N+2. The ones under 2000 are 20, 104, 464, 650, and 1952, all of which are primitive abundant as well. Barely Deficient: A number N whose aliquot sum equals N-2. The only ones under 2000 are 3, 10, and 136. and for a silly one: Perfectly Scrambled: A number whose aliquot sum is an anagram of itself. All perfect numbers are trivially perfectly scrambled, and the only other ones I found under 1000 are 411 and 604, with aliquot sums of 141 and 460 respectively. (note that these were all found by me manually looking through a list of aliquot sums rather than by using a computer to search for them, so I might have missed some)
It's worth noting that Perfectly Scrambled is base-dependent. If you used a different number base, then the list of Perfectly Scrambled numbers would be different.
i love how many people in comments engage with the math, but i can't engage too deeply with it. I just enjoy listening to you talk, it's very brain-aligning.
i always look forward to your videos. you're so fun and i like the clarity with which you teach concepts - i appreciate that you don't skip steps and are very explicit when you introduce a new idea or show a step-by-step process. also you're just a delight to listen to. also i loved the "one" counter - you're very funny. love your videos, you may be my favorite math content creator
Maybe because you didn't subscribe? It doesn't always send notifications when you don't subscribe. Also make the settings "all" and not "personalized" or "none".
I love this channel for how effectively it captures the joys and beauties of math without becoming suffocatingly academic or high-level. Other videos in SoMEpi are like "Here's how to factorize these functions in a weird way," "Here's what you can do with higher-dimensional math," "Look at this cool high-level maths theorem that involves calculus!" And then this channel is all about the simpler stuff like emergent properties of numbers themselves, or polyhedral properties. It's not less mathy for it, but it is more... playful. It's the kind of math you'd discover for yourself, rather than having it taught to you.
besides how every power of 2 is an almost perfect number, there is another interesting pattern regarding perfect powers and aliquot sums that I don’t often see talked about. Namely, the aliquot sum of any power of 3 will be (n/2)-1/2. See how the aliquot sum of 3 is 1, 9 is 4, 27 is 13, 81 is 40, and so on. Or, put another way, the aliquot sum of a power of 3 is always half of itself, rounded down to the nearest whole number
I was literally looking for a video just like this. I saw a post the other day asking "what three numbers sum and multiply to the same value?" And immediately i thought "well it's 1, 2, and 3 that they want, but I wonder if there are any sort of non-integer answers to this question."
20:13 Have you also played with quasi solitary and quasi friendly numbers? The obvious case is that all the primes would form an infinite club with quasi index 1, but the other figures' patterns could change a lot. Plenty of fun in this video and the first time I've seen log(log(n)) scaling. Lol
is there any number whose number of step in it's aliquot sequence to reach a prime/perfet/amicable/sociable number is itself a perfect number or itself ?
@@burner555 I know, im just saying that cause kuvina is enby (non-binary), and that makes sense. That yeah, i know that, the rainbow existed way before any queer symbol, way before humanity actually lmao xD But anyways, i get what you're saying, and also... Dont think you're being hateful, or a bigot. You're saying facts and truths, so dont be afraid to stand to your facts! Cheers, hope you have a nide day!
In the section “Quasi perfect” ( 6:06 ) you defined a quasi perfect number as s(n) = n - 1, but in the section “Almost perfect” ( 8:11 ) you defined quasi perfect numbers as s(n) = n + 1. Which one is it?
5:30 technically mercenne primes always have a prime exponent so it wouldn't be a mercenne prime. I literally mean that 2 ** k - 1 is *never* a prime if k is not and can only *possibily* be a prime if k is prime.
Kuvina is Anti Jan-Misali. Black Sans Serif on a White Background with plenty color Instead of White Sans Serif on a Black Background with little to no color
another fun fact about 70: on top of being a weird number, its also the smallest abundant number that's divisible by neither 4 nor 6. ofc any multiple of 6 is automatically abundant, and while multiples of 4 can be deficient they have a pretty high chance of turning out to be abundant, so its pretty rare, especially among 2 or 3 digit numbers, to see an abundant that has neither as a factor. 70 is the first; the second and third are unsurprisingly 350 and 490; multiples of 70. im not sure yet if 770 is the fourth or if there's one or more in between.
There are actually 2 in between, 550 and 650. The first few are: 70, 350, 490, 550, 650, 770, 910, 945. Then 88 of them have 4 digits, 830 have 5 digits, and 8502 have 6 digits. Seems like a solid 9/1000 numbers have this property.
@@redpepper74 thank you. unsurprising that theyre almost all multiples of 10. also interesting how theres several multiples of 50 here, and then they just stop: 850 and 950 are both deficient. and yes, 945 is quite literally the odd one out here.
technically they're defined as abundant numbers where you subtract one of their factors from the aliquot sum to get n. With powers of 2, you have to add a factor (1) a second time to get n
If there are even perfect numbers for every mersenne prime and we know primes are infinite(and I believe that there are also infinitely many mersenne primes), wouldnt we know that there are infinite perfect numbers(at least even ones)?
Is this a real question? Because if so, there is a discredited theory that the leader of a pack of wolves is the "alpha" of the pack, so someone decided to apply that to humans and call them an "alpha male" and from that spawned beta males, which are considered "lesser" to alphas, and sigmas, which are like alphas but more independent. This is all nonsense pushed by charlatans to sell online courses
1:23 wow wow wowowowo2owowobwow i didnt know that wow just wo wtf wow i mean wow i mean yeah but i mean yeah but also how,are there more aside from these?
BREAKING NEWS
After 6 years, a 52nd Mersenne prime has been found, which means there is also a 52nd perfect number!
I hope you like numbers because this video is extremely mathy! Thank for the patience awaiting the new video as I've been busy irl. I hope you enjoy!
is there a tl;dr for this
tl;dr numbers with funny properties
@@Kuvina a little bit more longer
Heck yeah! Numbers! :D
(Of course I love numbers, why do I think I’m subscribed to this channel??)
@@Kuvinanumbebbesbrs
Parker Perfect Numbers
Parker odd perfect numbers are actually even perfect number
YES
Leave him alone already
Omg you're so right. That, like, the funniest math joke I know and I'm actually sad that I see it so so rarely
booooo get new material
I once saw someone writing a code to determine if n was a perfect number. The code computed σ(n) by checking all numbers d up to the square root of n, and adding d and n/d to a total whenever d divided n.
However, when n=N² is a perfect square, the divisor d=N was not included in the sum (due to a < sign), and instead of comparing 2N² with σ(N²), the code was comparing it with σ(N²)-N.
I coined the false positives that the code may yield (which are a very niche subset of the near-perfect numbers) as PSEUDOPERFECT numbers.
I told the person who wrote the code that it was flawed. However, I was unable to find a counter-example. Over the years, I have checked up to n=458,335,615,276,564,171,975,521 (inclusive) without finding a single pseudoperfect number, but I can't discard that they exist.
I would love to know whether they exist, because even though it's been almost 10 years, if it turns out that pseudoperfect numbers don't exist, then the code would be valid and I should apologize to that person.
That's actually exactly how my own code works! Well except for the fact that I preemptively realized not double count sqrt(n) in those cases.
I did some work on this problem. It’s well known that square numbers can’t be perfect numbers, so any square returned by the algorithm would be wrong. This is because you get (even) 2N^2 = (odd) σ(N^2). For the algorithm to return a false result, we need 2N^2 = σ(N^2) - N, which would require N to be odd to make the whole expression even. Also, we can rearrange to get 2N^2 + N = σ(N^2), or N(2N+1) = σ(N^2) for some odd N. This tells us that N | σ(N^2) and 2N+1 | σ(N^2). This last statement may lead to a contradiction, proving the algorithm always works, but my number theory is rusty so I’ll have to stop here
I love how the ending "bye!" was timed and in-tune
Ok, the part about Sublime numbers actually blew my mind. I have a newfound appreciation for 12 and its Sublime sibling.
12 is a great number
Some other ones I’ve came up with (others probably have found these as well):
Barely Abundant: A number N whose aliquot sum equals N+2. The ones under 2000 are 20, 104, 464, 650, and 1952, all of which are primitive abundant as well.
Barely Deficient: A number N whose aliquot sum equals N-2. The only ones under 2000 are 3, 10, and 136.
and for a silly one:
Perfectly Scrambled: A number whose aliquot sum is an anagram of itself. All perfect numbers are trivially perfectly scrambled, and the only other ones I found under 1000 are 411 and 604, with aliquot sums of 141 and 460 respectively.
(note that these were all found by me manually looking through a list of aliquot sums rather than by using a computer to search for them, so I might have missed some)
That's so cool! And I like the names
It's worth noting that Perfectly Scrambled is base-dependent. If you used a different number base, then the list of Perfectly Scrambled numbers would be different.
That sublime number in the end was the most interesting piece of information in this video to me.
The Aliquot sequence, and how 276 seems to diverge, reminds me of the Collatz Conjecture...
same
I'd like for 22021 to be prime but unfortunately 19 is my favorite number and I cannot allow it to get removed from existence
i love how many people in comments engage with the math, but i can't engage too deeply with it. I just enjoy listening to you talk, it's very brain-aligning.
Always love your videos. Very high quality and a lot of passion and love is put into them. Thanks for sharing your passion with us other math lovers.
I noticed this video's length is perfectly round... (:
i always look forward to your videos. you're so fun and i like the clarity with which you teach concepts - i appreciate that you don't skip steps and are very explicit when you introduce a new idea or show a step-by-step process. also you're just a delight to listen to. also i loved the "one" counter - you're very funny. love your videos, you may be my favorite math content creator
thank you! you're the first one to leave a comment about the 1 counter!
@@Kuvina WHAT! no way! i can't believe that. no way at all. NO ONE has mentioned it??? that's maybe one of my favorite things LMFAO
Why didn't UA-cam send me a notification about a video by one of my favourite creators??
did you hit the bell icon
Make sure the notifications are on “all” instead of “personalized”
Maybe because you didn't subscribe? It doesn't always send notifications when you don't subscribe. Also make the settings "all" and not "personalized" or "none".
This year(2024) is actually a Quasi aliquat perfect number (see 15:08)
I love this channel for how effectively it captures the joys and beauties of math without becoming suffocatingly academic or high-level. Other videos in SoMEpi are like "Here's how to factorize these functions in a weird way," "Here's what you can do with higher-dimensional math," "Look at this cool high-level maths theorem that involves calculus!" And then this channel is all about the simpler stuff like emergent properties of numbers themselves, or polyhedral properties.
It's not less mathy for it, but it is more... playful. It's the kind of math you'd discover for yourself, rather than having it taught to you.
Collatz conjecture flashbacks
"3x+1" "stop it Patrick you're scaring him"
2:23 oh no you have summoned the gen alpha kids
Fr
So help me, if I see any "skibbidi toilet" numbers, there's gonna be a revolutionary advancement in war crimes.
would try to send them into the imaginary realm
Imagine:
Womp womp numbers
Gigachad numbers
Based numbers
Fries in the bag numbers
Lil bro numbers
Alpha numbers
Gyatt numbers
Rizz numbers
Ohio numbers
Slay numbers
Preppy numbers
Oiled up numbers
Caked up numbers
Clapping numbers
Mewing numbers
@@skippitysmithsonshorts NAH XDXDXD
very educational
or not
now im just filled with next to useless information about imperfect numbers
not in a bad way, i love the video :)
besides how every power of 2 is an almost perfect number, there is another interesting pattern regarding perfect powers and aliquot sums that I don’t often see talked about. Namely, the aliquot sum of any power of 3 will be (n/2)-1/2. See how the aliquot sum of 3 is 1, 9 is 4, 27 is 13, 81 is 40, and so on. Or, put another way, the aliquot sum of a power of 3 is always half of itself, rounded down to the nearest whole number
I've done this recently, ignoring 1 as a prime, and have come up with weird things, and found out about betrothed numbers in that adventure.
Which of these types of numbers do you like the best?
multi perfect!
I was literally looking for a video just like this. I saw a post the other day asking "what three numbers sum and multiply to the same value?" And immediately i thought "well it's 1, 2, and 3 that they want, but I wonder if there are any sort of non-integer answers to this question."
It feels odd that we are stuck on the 276 aliquot sequence, with modern computing it feels like we should just be able to crank that out
3:20 The brainrotted will only notice sigma.
your videos are very relaxing
it has onnly 2007 view it deserves more
now 2015
I’d like to say I understand all of this but, my brain exploded trying to understand it XD.
GUYS YOU KNOW THE RULE, IF KUVINA VIDEO WE EMIDIATELY WATCH!!!
0:15 the 8th: 2.31 quintillion
the 9th: 2.66 undecillion
‘the sigma function’
**sighs**
**opens comments**
Sigma is a greek letter, not ur brainrot version
This video is almost perfect.
oh yeah, i remember the WILD RIDE that was that numberphile video
I actually found a Unitary Sociable Loop of 3 (30,42,54) and 2 Unitary Aspiring Numbers before reaching the Unitary Perfect Number 90 (66,78,90)
HE HEARTED MY COMMENT! Also, 100 is the only number between 1-100 that is socially aspiring (100,30,42,54)
@@mrhangertv1829kuvina uses they/them
The perfect video.... 30 minutes exact
Amicable numbers are my favorite
Hey mom wake up, new kuvina video dropped
I wish if there a number that is perfect in all these ways combined
I wonder if the OEIS has a name for the sociably aspiring numbers
... but I still prefer 37.
20:13 Have you also played with quasi solitary and quasi friendly numbers? The obvious case is that all the primes would form an infinite club with quasi index 1, but the other figures' patterns could change a lot.
Plenty of fun in this video and the first time I've seen log(log(n)) scaling. Lol
No way Kuvina uploaded!
The Archimedean perfect numbers
The negatives will be called the Catalans
from 28:10 it sounds like an illuminati presence proof
is there any number whose number of step in it's aliquot sequence to reach a prime/perfet/amicable/sociable number is itself a perfect number or itself ?
New Kuvina Saydaki video, life finally has a meaning
Here before Gen Alpha starts joking about the sigma function
gen alpha arent going to be here theyre like 11 years old they dont care
@@NocturnalTyphlosion almost forgot that gen alpha can’t read. Thanks for the reminder
I feel so σ!
I love your videos so much
the colours are always arranged into the lgbt flag sequence, awesome
Sorry to burst your bubble, but rainbows have been arranged like this way before the become a queer symbol
@@burner555 I know, im just saying that cause kuvina is enby (non-binary), and that makes sense. That yeah, i know that, the rainbow existed way before any queer symbol, way before humanity actually lmao xD
But anyways, i get what you're saying, and also... Dont think you're being hateful, or a bigot. You're saying facts and truths, so dont be afraid to stand to your facts!
Cheers, hope you have a nide day!
Great Video. Thank you
In the section “Quasi perfect” ( 6:06 ) you defined a quasi perfect number as s(n) = n - 1, but in the section “Almost perfect” ( 8:11 ) you defined quasi perfect numbers as s(n) = n + 1. Which one is it?
Quasi perfect numbers are s(n)=n+1. They can alternatively be defined as n=s(n)-1, which is how I define them in the first section
gg kuvina is back
aspiring infinitism
there would technically be an infinite number of numbers that rise up to infinity if 276 is proven to rise up to infinity
When is the next relativity video?
Please please slow down and make separate videos for each kind of number. Otherwise u are almost perfect❤️👏
5:30 technically mercenne primes always have a prime exponent so it wouldn't be a mercenne prime. I literally mean that 2 ** k - 1 is *never* a prime if k is not and can only *possibily* be a prime if k is prime.
Kuvina is Anti Jan-Misali.
Black Sans Serif on a White Background with plenty color
Instead of White Sans Serif on a Black Background with little to no color
another fun fact about 70: on top of being a weird number, its also the smallest abundant number that's divisible by neither 4 nor 6. ofc any multiple of 6 is automatically abundant, and while multiples of 4 can be deficient they have a pretty high chance of turning out to be abundant, so its pretty rare, especially among 2 or 3 digit numbers, to see an abundant that has neither as a factor. 70 is the first; the second and third are unsurprisingly 350 and 490; multiples of 70. im not sure yet if 770 is the fourth or if there's one or more in between.
There are actually 2 in between, 550 and 650.
The first few are: 70, 350, 490, 550, 650, 770, 910, 945.
Then 88 of them have 4 digits, 830 have 5 digits, and 8502 have 6 digits.
Seems like a solid 9/1000 numbers have this property.
@@redpepper74 thank you. unsurprising that theyre almost all multiples of 10. also interesting how theres several multiples of 50 here, and then they just stop: 850 and 950 are both deficient. and yes, 945 is quite literally the odd one out here.
2:23 is sussy
10:38 28 does not want to be with anyone else
what about: antiperfect numbers. aka primes
I will make it my life mission to find 10 a friend
it´s some math
Seeing them say sigma hurts me.
Oh why? Cuz its brainrot? If you think it's brainrot, then YOU are brainrot. Kids these days
@@NotLobotomy I know that in this case it's not related to brainrot, but it still hurts me
1:52 sorry the... what project???
Gimps, not goons, brainrot being.
sigma is multiplicative, but also sussy...
7:37 why does the number have to be even? The 2 can have any exponent, but anything greater than 0 would make it even?
All powers of 2 are also near-perfect numbers (just one off), but that would be too easy.
technically they're defined as abundant numbers where you subtract one of their factors from the aliquot sum to get n. With powers of 2, you have to add a factor (1) a second time to get n
Actually, those numbers are deficient so they can't be Near Perfect
If there are even perfect numbers for every mersenne prime and we know primes are infinite(and I believe that there are also infinitely many mersenne primes), wouldnt we know that there are infinite perfect numbers(at least even ones)?
I don't believe we know that Mersenne primes are infinite
It is unproven that there are infinite mersenne primes
Interesting. Why then would we be using mersenne primes as our main search for larger primes?
@@k0pstl939 because it is easy to prove if a mersenne number is prime, also it is suspected but unproven that there are infinite mersenne primes
@@k0pstl939it's also unproven that there *aren't*, we just don't know currently
Gen Alpha ruined maths for me. I will never hear "Sigma" the same way again
i was expecting a top comment to be "sigma function more like me function" or something
Can someone explain to me the aliquot thing
Huh. I still don't know why my brother keeps on saying he's a sigma.
Is this a real question? Because if so, there is a discredited theory that the leader of a pack of wolves is the "alpha" of the pack, so someone decided to apply that to humans and call them an "alpha male" and from that spawned beta males, which are considered "lesser" to alphas, and sigmas, which are like alphas but more independent. This is all nonsense pushed by charlatans to sell online courses
Omg perfect numbers
HOW AM I HERE IN AN HOUR
I found a quasi-perfect number
Are you Fermat?
sigma 💀
ok
Username say it all
this is for real math class number g64
nice
1:23 wow wow wowowowo2owowobwow i didnt know that wow just wo wtf wow i mean wow i mean yeah but i mean yeah but also how,are there more aside from these?
Cool!
Maybe 138 goes to the odd perfect number 😄
Love how one is just in it different category just like it in a different category for prime or composite numbers it’s just 0,1
1:04 what about 69
hi kuvina! lovely video. is there a place to get "news" about new discoveries of number facts like this?
almost, near, quasi. is it a maths video or a synonym dictionary?
a.. thesaurus?
Is any almost perfect number odd at least? 😭
Edit: YEAH BOYS 1 IS ALMOST PERFECT!!! 🎉🎉🎉
nice flag 🟨⬜️🟪⬛️
i suppose you could say they dont have enough sigma rizz to be perfectg
god save you
22021 why did you have to be not prime 😭
ALMOST r/foundsatan
Id like for 9000 and 5397 to be coprime but unfortunately they share a common factor of 3
Hi
Okay but what is a perfect number supposed to be
Oops I watched like 10 seconds in and now I think I know
A perfect and almost perfect video 🤔
the friggin sigma function wtf
XD
68
Sigma
i forgot 1 can be multiplied by 1