The Song Fight Bulletin Board

Question: What is the lowest positive integer that has never been spoken by anyone out loud in any language in the history of the world? I'm thinking it's around 10 million or so. My guess is 11,207,819.

... and probably as soon as somebody read that, they spoke it aloud just to mess with you (i didn't).
although i'd expect it's much lower, like maybe a negative number
-bill

Hmmm. yeah, the lowest positive integer, I should have said. Wait, I DID say that. (Check edit)

Man, that is an ugly dog - I'd put the poor thing down

Yes, but what I really wonder is what's the lowest positive integer that has never been spoken out loud by anyone in any language in the history of the world.

Have any of you ever said something and then wondered whether you're the first person in the history of the world to ever say that exact thing in those exact words? When I think that no, I'm the first to say that, it gives me a little thrill. I get them where I can.

Possibly the first person in the history of the world to ever say that exact thing in those exact words and not follow it with 'My that was a stupid question. Sorry to waste your time, everybody.' :p

Glad you're still around to detest me. I come back and things haven't changed!

Heheheh - well, detest is a little strong, but I am nothing if not predictable!

I'm betting that various groups of teenagers over the years have counted, outloud, far beyond 11 million. I once dug to China in my backyard, after all.

Now I'm doing some math and finding that would take 7 years if counting three integers a minute, 365/24/7.

But then there are all those thousands of scientific results that have occurred since the beginning of time where someone actually read the number outloud. Surely there aren't many holes left?

So, yeah, waste of time.

Yeah, not so much how high has someone counted, more like what you said about scientists and smart people who deal in high numbers. No matter what, there's SOME number that's never been said.

Waste of time? Well, I don't think so... wait a sec... hey, yeah. I guess it is.

Lets test if Jim's 10,000,000-ish number is plausible. Lets assume (for convenience) that the number who's name hath not been spoken is less than 10,000,000. Now lets take the set of all psuedo-random non-unique utterances of numbers between 1 and 10,000,000. Specifically, I'm only really talking about large, essentially random numbers, like those a scientist or math teacher might read aloud. Let's assume these are totally random, even though they probably aren't (I'm sure "10 million dollars" gets said a lot more frequently than "7,014,883"). Now let's figure out how big this set would have to be to give us even odds that all numbers between 1 and 10,000,000 have been uttered.

• Let X be the highest number we are considering (in this case 10^7).
  Let Y be the number of times these large numbers have been uttered.

After Y utterances of numbers less than X, the likelihood that any given number in this range has been uttered is P = 1 - (1/2)^(y/x). Therefore the likelihood that EVERY number in the range has been uttered is P^x. This yields a formula for computing the probability that if, after Y guesses from a set of X things, all X things have been chosen: f(x, y) = (1 - (1/2)^(y/x))^x.

So, to find the answer for the italicized problem in the first paragraph, we need to solve f(x, y) = 0.5 for y. I wont bore you with the math--chances are I have already--but solving for y gives y = log_2(x^x). In this case, the number of times random numbers between 1 and 10,000,000 must be spoken to give even odds that every one has been said is log_2(10,000,000 ^ 10,000,000), or about 230 million times. Is this number reasonable? Not really. I'm sure that nearly every adult in the US has uttered a random 7 digit number at least once in their lifetime, and that's 250 million utterances right there.

So Jim, I think the number you're looking for is much higher. If someone can come up with a reasonable estimate for the number of utterances that have occurred they could plug it into this formula and get some kind of answer.

http://www.onebillionmazes.com/?p=11207819
Note the Puzzle number.

http://www.ncbi.nlm.nih.gov/entrez/quer ... t=Abstract
Note the PMID of the article.

http://www.alibaba.com/catalog/11207819 ... Boxes.html
Note the URL.

http://ldas-cit.ligo.caltech.edu/ldas_o ... .820509964
note the number of files scanned on line 96.