Can someone ELI5 this post?
Original response:
Almost correct, but it's 2^49081-1, not (10^49081-1)/9. And there's no division by 9 at all.
You know how some things are red and some are not? There are probably multiple red things in the same room as you are right now. Most people can easily tell if something is red or not by simply looking at it.
But what about places we cannot see? Even though we have never been there, it is a save bet that there are red things in let's say New York. And if we wanted to make sure, we could book a flight and go check. But is there something red in every village in the world? My guess would be yes, but we do not know for sure and it will be hard to visit every place because there are so many.
At this point you might have explained some things, but if you have not lost them yet you could take this further.
There are places even further away than every village. We know that Mars is red because it is so big and people have sent a robot there to be absolutely sure. But what about other planets? It is hard to tell because planets can be very far away and red things can be very small. And going there is no option because even the fastest rocket would not get there any time soon.
What this post mentions is the equivalent of someone building a very precise telescope fixed at a very specific place very far away and they saw something red there. Here the analogy breaks because there are other more conceptual problems with building such a telescope. Otherwise, I am surprised how many similarities there are actually :)
Numberblocks that are rectangles are not prime.
My son knew of "primality" at 5 because of this.
My 4.5yo will have no chance, more interested in sharks, trains or even better shark trains!
> As of March 2022, the largest known prime number 282,589,933 − 1, the largest probable prime R8177207 and the largest elliptic curve primality prime R49081 are all repunits.
const R49081 = BigInt('1'.repeat(49081));https://en.wikipedia.org/wiki/Repunit
It's part of "recreational mathematics" which solves a longstanding question I've had after finding out about a "special" set of numbers that make me think "What the hell was that?"
Basically the question was "Why can't I just come up with any dumb rule I want? Like the number of letters in the number being equal to the number of digits in the base ten representation or some other noise like the number of toothpicks required to represent it in decimal and then do some wacky proofs regarding "toothpick count"?" Apparently the answer is "well you can!"
Alright then. There exists a non-serious branch of math.
This remains an unproven conjecture that many hope to be true . . .
Although I will concede those are not mutually exclusive claims.
You might not be happy that someone investigated this because you apparently made up this example in order to make fun of this kind of question, but I looked into this a little bit and it's more subtle than I first thought.
The typical random integer small enough to have a well-defined English name (following Guy and Conway's prefixes at https://en.wikipedia.org/wiki/Names_of_large_numbers#Extensi... which will go up to 10³⁰⁰⁰-1) has a significantly longer English name than its decimal expansion. E.g. if we take fifty random digits like
96689907044105050355231217362382284051155144740067
the name of this number is "ninetysixquindecillionsixhundredeightyninequattuordecillionninehundredseventredecillionfortyfourduodecilliononehundredfiveundecillionfiftydecillionthreehundredfiftyfivenonilliontwohundredthirtyoneoctilliontwohundredseventeenseptillionthreehundredsixtytwosextillionthreehundredeightytwoquintilliontwohundredeightyfourquadrillionfiftyonetrilliononehundredfiftyfivebilliononehundredfortyfourmillionsevenhundredfortythousandsixtyseven"
or 430 letters, while the number is written with only 50 digits.
For small numbers, the closest approach between the lengths of the two is 10 (len("ten") == 3, len("10") == 2). And the typical word length grows faster than the digit length.
But the lengths do cross over many times up in the illions, as the illion terminology is unusually short compared to the size of the number described. Specifically, the smallest integer where this occurs is 1,000,000,000, where len("onebillion") == len("1000000000"). That's the smallest member of your set.
There are other examples such as 10¹³+1, 10¹³+2, 10¹³+6:
len("tentrillionone") == len("tentrilliontwo") == len("tentrillionsix") == len("10000000000001")
And there are many more further out into the illions.
For any example in which "one", "two", or "six" appears in the name of a number, any of the others can be substituted and the property will still hold. Likewise for "four"-"five"-"nine" and "three"-"seven"-"eight".
Here's another one for you. Could some uncountable infinities be cardinally smaller than countable infinites if we throw out the property that they must be countable by definition? As in you can say "this is smaller than that, I can demonstrate it's infinite but I can't do a countable demonstration."
No, trivially. If X has a smaller cardinality than Y then there is an injective function from X to Y (by definition), if Y is countable then there is an injective function from Y to the natural numbers (by definition), then by composing those two functions you have an injective function from X to the natural numbers i.e. X is countable.
EDIT: the reference
https://en.m.wikipedia.org/wiki/The_Nine_Billion_Names_of_Go...
Try paging through OEIS sometime... https://oeis.org/
There are many seemingly silly bits of math. However, the answer in mathematics is often not the most interesting part (or even interesting at all), and a lot of important mathematics was created to produce answers for seemingly silly or pointless questions.
If you have a silly math problem and it's answered quickly-- well then it didn't waste much time. If you have a silly math problem and it's fiendishly hard to solve, then the discoveries that lead to the solution aren't silly-- they teach us how to do something we didn't know how to do before and may have important and useful implications.
For things like these primarily proofs the mathematical discoveries have already happened (at least until someone comes up with a new one!)-- and the ongoing effort is in engineering for numerical software and the IT work required to manage the computation in a cost effective manner.
Some families of primes have important mathematical structure that has a lot of known consequences, special algorithms that work with them (such as repunit primes in base 2), etc. Other's don't, or at least don't have any that we know of yet-- but they still divide up the literally infinite space of primes and give people interested in studying them a subset they can focus on.
So, for this reason, a repeating sequence of 1s is a natural choice to try and find prime numbers.
Present me wishes I could explain Explain ELI5 Like I'm 5 to my past self.