Even numbers are great, but have you ever heard of threeven numbers?
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The even numbers are the ones divisible by two, and the odd numbers are everything else. Two is famously the only even prime, but that isn’t some deep coincidence. Every other even number has two as a factor, on top of one and itself, so it can’t be prime. The even numbers are disqualified by definition.
Now do the same thing with three. Call the numbers divisible by three “threeven” and everything else “throdd.” Three is the only threeven prime, for exactly the same reason two is the only even prime. But here’s where three is more interesting than two: there are two different kinds of throdd numbers. Some numbers are one more than a multiple of three (1, 4, 7, 10, …). Call those throdd up. Others are one less than a multiple of three (2, 5, 8, 11, …). Call those throdd down.
So which kind do the primes prefer? Number theory says that in the long run, neither: the primes should be evenly distributed among the threeven and throdd prime numbers. But start counting and keep score, and something strange happens. Throdd down takes the lead almost immediately and refuses to give it up. Sometimes the lead shrinks to a single prime. But it’s remarkably consistent. Throdd up doesn’t pull ahead until 608,981,813,029.
This lopsided race has a name, Chebyshev’s bias, after the 19th-century Russian mathematician who noticed the same thing in a different race: primes one less than a multiple of four tend to outnumber primes one more than a multiple of four.
So why does throdd down keep winning? Look at what happens when you square these numbers. A threeven number squared is still threeven, much like an even number squared is still even. A throdd number squared stays throdd. But it’s more specific than that. A throdd up number squared is throdd up, and a throdd down number squared is ALSO throdd up. The two “minus ones” that make a throdd down number act like negative one times negative one, and they cancel into a plus one.
That matters because, roughly speaking, the theory says it’s the primes together with their squares that should split evenly between the two sides. All of those prime squares sit on the throdd up side. Set them aside and count only the primes, and throdd up is the side that comes up short. The bias isn’t a fluke. It’s a fingerprint left by the squares.
But the win isn’t permanent. Past 608 billion, the lead flips back and forth infinitely many times. It’s a good reminder that a pattern holding for the first hundred thousand million billion numbers still might not hold forever.
#primenumbers #numbertheory #chebyshevsbias
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Did anyone suggest oddd instead of throdd?
I can buy the explanation for why there's a bias for 3n+1 primes as opposed 3n-1 primes, but then why does the count ever swing back to favor the 3n-1 primes?
YouTube auto captions are getting really good
we just making up words now
Throdd.
Help. I'm having a stroke.
Is… this a homestuck reference..?
This last sentence is exactly the same reason why the Goldbach Conjecture remains unproven. Just because you identify a pattern doesn't mean you can assume that pattern will always hold to infinity.
That last bit about "…that doesn't mean that whatever pattern you've noticed will go on forever" is a good way to look at shitty life situations, too. There is always a possibility for change.
as a dyscalculic, if i watch more than 10 seconds of this video, i might have a heart attack. but thank you for the content anyway
Is "threeven" and "throdd" real mathematical terms?
Chebyshev Point? Good ol' Pafnuty at it again! Is there a field of Mathematics he didn't work on!?
this is interesting. and i want to see fifodd and fifeven now!
How do you even being to figure this out
I thought about the existence of throd numbers when I was like 8
this is like a The grey Cuber bit
Throdd
This is why it’s so important to realize that simply knowing the Riemann hypothesis or collatz conjecture holds for very large numbers doesn’t mean we actually know they are true. Some patterns take a very long time to be broken but still break all the same
Throdd sounds like something that is frog and a tadpole.
My kid, the moment I lay down:
"1 throdd up."
Say what? A slowed down narration would be helpful here. Otherwise you lost after intro'ing "throdd" numbers.
23 is a thraud
Onodd: {0}
Oneven: all the rest
I had to make sure this wasn't an older video from April 1st.
That's cool that some form of counting primes has a kind of normal graph.
Funnily enough, though, if you pick a large enough number, and then a random number from 2 to that, chances are there will be more throdd down primes leading up to it than throdd up primes.
My head is throdding
Sounds a lot like apophenia 😂
How does throdd up ever catch up then??
I do this splitting of primes to have unique dimensionality coordinates stored. The numbers get ridiculous quickly but computers are good
i throdd up and i throdd down
Always since I was young I pondered what could be the mysterious reason why the universe and laws of logic chose 2 to be the only even prime. I never considered that question itself was so flawed, I can't believe I never realized this before.
took me back to 8th grade
Even numbers suck because the goated prime numbers (lonely numbers that can only be divided by one and themselves) are mostly all odd
NAH, Odd numbers are the best. All of my favorite numbers are odd numbers, like 25, 45, 55. Every number that ends with 5 is my favorite
It's actually really significant when a prime is 1 mod 4, vs 3 mod 4, for some important reason in number theory
Who up threeven they throd