>  What is the smallest number for which it's unknown whether it's prime
> or composite?

1000000000000000000000000000000000

Oh, wait, you just worked out that is composite in your head, so ermm I 
guess it's

1000000000000000000000000000000001

now :-)

BTW, how does everyone else know if such numbers have been tested or not, 
it's not like prime numbers where they are published and announced, people 
probably have done this without publishing the results, so it's just going 
to be some relatively small number that just happens to have never been 
tested by anyone.  Good luck finding out what it is :-)

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