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http://en.wikipedia.org/wiki/Repunit

But I see no proof there.

In[1]:= PrimeQ[1111111111111111111]
Out[1]:= True

Fermat's Little Theorem and proof by contradiction?

Seems unfeasible for a number so large, but if you have some hefty computing power . . .

I guess the conclusion is that there is only proof by computer (via some algorithm).