253 composite and 293 prime factors now,

10 years later.

As for the proof, I tried to travel back 150 years to the times of Thomas Clausen and Le Lasseur. They've

proven 11111111111111111 composite about the same way (they arguable knew the restricted residue classes, btu the rest is simply dividing away during long winter evenings). For 1111111111111111111, some gentleman divided away until he'd proven it prime; he submitted the works to the London Mathematical Society, and a specially appointed committee of that body accepted the proof as final and conclusive. See the Proceedings of the Society for 14th February, 1918.