3 ms·

Apparently it's just that number of 1s followed by each other as a number. https://en.wikipedia.org/wiki/Repunit https://en.wikipedia.org/wiki/Repunit It's pa

Apparently it's just that number of 1s followed by each other as a number. https://en.wikipedia.org/wiki/Repunit 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.

defrost 4y ago
> There exists a non-serious branch of math. This remains an unproven conjecture that many hope to be true . . .
- kristopolous 4y ago
  well this gives me hope that the solemn attitudes surrounding recreational mathematics is just math kayfabe and I'm not actually nuts. Although I will concede those are not mutually exclusive claims.
  - defrost 4y ago
    George Conway and others have at times lamented that whenever somebody arrives at something sufficiently complex to be intriguing and no apparent real world application ... military gentrification follows within a decade.
schoen 4y ago
> Like the number of letters in the number being equal to the number of digits in the base ten representation 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#Extensions_of_the_standard_dictionary_numbers 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".
- kristopolous 4y ago
  Thanks I'm pretty sure somebody had taken the time. 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."
  - lmm 4y ago
    > 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.
    - deleted 4y ago
      [deleted]
- wrycoder 4y ago
  I just noticed that there don’t seem to be as many stars out tonight as there usually are.
  - kickaha 4y ago
    I understood that reference! EDIT: the reference https://en.m.wikipedia.org/wiki/The_Nine_Billion_Names_of_God https://en.m.wikipedia.org/wiki/The_Nine_Billion_Names_of_Go...
nullc 4y ago
> There exists a non-serious branch of math. Try paging through OEIS sometime... https://oeis.org/ 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.
IIAOPSW 4y ago
Its not just any old dumb rule. Say I wanted to make something really unlikely to be divisible. Well, let's try taking something very large and very evenly divisible, and then add or subtract a 1 so that it's always just a tiny fraction off from being divisible. In the addition case we might get something like 10000001. In the subtraction case for the same number we would get 9999999, after which we could factor out 9 to get 1111111. You could also do it with numbers that don't line up with base 10. 360 is a highly divisible number (hence its use for angles) but 361 / 359 are going to be 1/360th out of step with those divisors (and in fact, 359 is prime). So, for this reason, a repeating sequence of 1s is a natural choice to try and find prime numbers.

defrost 4y ago

> There exists a non-serious branch of math. This remains an unproven conjecture that many hope to be true . . .

kristopolous 4y ago
well this gives me hope that the solemn attitudes surrounding recreational mathematics is just math kayfabe and I'm not actually nuts. Although I will concede those are not mutually exclusive claims.
- defrost 4y ago
  George Conway and others have at times lamented that whenever somebody arrives at something sufficiently complex to be intriguing and no apparent real world application ... military gentrification follows within a decade.

schoen 4y ago

> Like the number of letters in the number being equal to the number of digits in the base ten representation 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#Extensions_of_the_standard_dictionary_numbers 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".

kristopolous 4y ago
Thanks I'm pretty sure somebody had taken the time. 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."
- lmm 4y ago
  > 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.
  - deleted 4y ago
    [deleted]
wrycoder 4y ago
I just noticed that there don’t seem to be as many stars out tonight as there usually are.
- kickaha 4y ago
  I understood that reference! EDIT: the reference https://en.m.wikipedia.org/wiki/The_Nine_Billion_Names_of_God https://en.m.wikipedia.org/wiki/The_Nine_Billion_Names_of_Go...

nullc 4y ago

> There exists a non-serious branch of math. Try paging through OEIS sometime... https://oeis.org/ 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.

IIAOPSW 4y ago

Its not just any old dumb rule. Say I wanted to make something really unlikely to be divisible. Well, let's try taking something very large and very evenly divisible, and then add or subtract a 1 so that it's always just a tiny fraction off from being divisible. In the addition case we might get something like 10000001. In the subtraction case for the same number we would get 9999999, after which we could factor out 9 to get 1111111. You could also do it with numbers that don't line up with base 10. 360 is a highly divisible number (hence its use for angles) but 361 / 359 are going to be 1/360th out of step with those divisors (and in fact, 359 is prime). So, for this reason, a repeating sequence of 1s is a natural choice to try and find prime numbers.