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    $\begingroup$ Not quite relevant, but are you aware that it has never been proved that there are infinitely many Mersenne composites? $\endgroup$ Commented Dec 8, 2013 at 3:36
  • $\begingroup$ A similar question was raised on MO by Igor Rivin earlier: see mathoverflow.net/questions/81729/mersenne-quasi-primes $\endgroup$ Commented Dec 8, 2013 at 15:53