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Does there exist a property which is known to be satisfied by only one integer, but such that this property does not provide a means by which to compute this number? I am asking because this number could be unfathomably large.

I was reading Conjectures that have been disproved with extremely large counterexamples? , does there exist a conjecture that is known to have a counterexample, but which has not been found, and where there is no "bound" on the expected magnitude of this integer?

Is there known something about how the largest integer that is expressable in n symbols, grows with n?

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Sure, for silly reasons. Take any noncomputable function, e.g. the Busy beaver function BB(n). Then a number like BB(10) is the unique number which is BB(10), but that doesn't tell you how to (easily) compute it. –  Qiaochu Yuan Feb 25 '11 at 20:21
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i think this is interesting since you can usually at least bound quantities (skewe's number, ramsey numbers etc.) even if the bounds aren't good, but I can't think of an example to "there exists example of x but no bounds whatsoever... –  yoyo Feb 25 '11 at 21:03
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up vote 5 down vote accepted

One can easily generate "conjectures" with large counterexamples using Goodstein's theorem and related results. For example, if we conjecture that the Goodstein sequence $\rm\:4_k\:$ never converges to $0$ then the least counterexample is $\rm\ k = 3\ (2^{402653211}-1) \approx 10^{121210695}\:$. For much further discussion of Goodstein's theorem see my sci.math post of Dec 11 1995

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I am extremely impressed with your ability to find newsgroup postings from 15 years ago. How do you do this? –  Harry Stern Feb 25 '11 at 20:37
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@Harry: To search the Google Groups Usenet archive go to groups.google.com and submit queries via "Search groups". For above use the query: $\ $ group:*math* author:dubuque goodstein. Or use the URL groups.google.com/groups/… –  Bill Dubuque Feb 25 '11 at 21:08
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@Harry Stern: It was his own post. :) –  ypercube Feb 26 '11 at 0:27
    
@ypercube, I know, that's why I'm impressed. I hope I'm that organized in the future. –  Harry Stern Feb 26 '11 at 8:54
    
and @Bill A search makes somewhat more sense than having some sort of catalog of all your posts, I guess, but still. That's nearly my entire lifetime :P –  Harry Stern Feb 26 '11 at 8:56
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