Smallest known unfactored composite number? I'm trying to find examples of "small" numbers which are known to be composite, but for which no prime factors are known.  According to this website the number $109!+1$ is a composite number of 177 digits, but no factors are known.  However, I can't find anything more up-to-date; maybe that number has been factored now; maybe there is a smaller unfactored composite number.
Anyway: does anybody know the smallest known composite number for which no prime factors are known?
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Addendum. With further exploration of the above pages I've found that the Wolstenholme number which is the numerator of
$$\sum_{k=1}^{163}\frac{1}{k^2}$$
has 138 digits and is composite, and no factors are known, as of July 16, 2012.  This is the smallest such number I've found so far.
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More: In the most recent (third) edition of the book of factorization of Cunningham numbers ($b^n\pm 1$) by Brillhart et al, the number $2^{1462}+1$ includes in its factorization a 130-digit composite number which at the time of publication had not been factored.
 A: Here's a number of 50 digits:
$$84286144766718574585896327097775856948442086719729$$ 
It's a good bet that nobody has yet considered this particular number (just because there are so many 50-digit numbers, and this one was chosen randomly).
Maple says it's composite, but since nobody else has considered this number and I don't know the factors, it's an example of a number that is "known to be composite, but for which no prime factors are known."
Oops: now I do know them: $178601959352247480503$ and $471921725116606004970765902743$.
But you get the idea...
A: There are a huge number of composite numbers that have unknown factors, most because nobody has tried.  I'll bet nobody had factored $20754285234059597221$ before I just tried PrimeQ(20754285234059597221) at Alpha.  Unfortunately, Alpha not only told me it isn't prime, it factored it as $22892731 \times 906588437791$.  I found it by typing a bunch of numbers on the keyboard, then appending seven zeros and finding the next number smaller that was $1 \pmod {19!!}$  I would just take some number with $20$ digits, find a near one that is $1 \pmod {19!!}$ and try the Fermat test at increments of $19!!$ until you find one that is composite.  I'll bet it has never been factored (but could be easily).  The $19!!$ guarantees it does not have a factor below $19$.
A: I'm going to be a little more explicit about the problem here, which other answers have only alluded to. The question you asked cannot be definitively answered, because it's related to something called the "interesting number paradox". Is there a smallest uninteresting number? If we found out what it was, wouldn't that suddenly make it "interesting "? The analogy isn't a complete match, but it gets us in the ballpark.
Was there a smallest unfactored composite number at the time you posted your question? Sure. Did we know what it was? Probably not. But even if we did, someone would have soon come along and factored it. 
Barring some special cases, smaller numbers are generally easier to factor, especially since we can just let a computer do it for us.  Once we know the factors of all the numbers below a particular one, that number's factorization will quickly fall to a sieve method. $345654323456756789365772498641397346521$ took Wolfram Alpha all of 5 seconds to factor.
Is there a smallest unfactored composite number right now? Yes, but only because no-one's gotten around to factoring it.  I doubt any such number, once known, could hold the title for longer than a day. 
(Maybe we could call this the "Big Yellow Taxi Paradox", because of the story of Ramanujan and taxi number 1729, and also the refrain of the Joni Mitchell song.)
