Seeking Prime Repunits

Fermat's Little Theorem states that if m is a prime then b^m = b 
(mod m) for every integer b.  Therefore, if we can find an integer b 
for which b^m is NOT congruent to b modulo m, we will have proven that 
m is composite.  Take, for example, b = 2, and set m = (10^71-1)/9.  
If m is a prime we would find that 2^m = 2 (mod m).  However, a little 
calculation shows that in fact 2^m is congruent to

8422611138482701447440679341973903379774416284229892522771978154015187

modulo m, so it follows that m is composite.  It helps to have a multi-
precision calculator or programming package to perform calculations of 
this kind.  If you have UBASIC, you can test for compositeness of 
base-10 repunits using the following little program.  (Note: UBASIC 
uses the "@" symbol for "modulo".)

10    INPUT "enter p"; p
20    m = (10 ^ p - 1) \ 9
30    b = 2 : q = 1
40    FOR j = 1 TO p
50      qt = q
60      FOR k = 2 TO 10
70         q = (q * qt)@m
80      NEXT k
90      q = (q * b)@m
100     PRINT j, q
110   NEXT j

The only values of p < 10000 that will give a final output of 2 
are 2, 19, 23, 317, and 1031.  This just means that they are "probably"
primes, because composites that satisfy this test are extremely rare.
However, it has been proven by other methods that these repunits are
in fact primes.

For a long time those five numbers were the only known prime repunits 
(in decimal), but in September 1999 Harvey Dubner found the next 
"probable prime" repunit with p = 49081.  This most likely is a prime, 
but it would be prohibitive (by any known methods) to give a 
deterministic proof.