(In reply to

possible solution by broll)

I had just come up with the same proof, broll, and was about to write it up. Thanks for saving me the trouble, broll.

Not sure why you single out N = 0. Lots of numbers only require 1 digit. Any number that does not have a factor of 2 or 5 only needs 1 digit. Also, 2 and 4 and 5 and 6 and 8 times any repunit only needs one digit.

*(Of course, if we allow M = 0, then this problem is difficulty 1, and it is true for rational, irrational, transcendental and imaginary values of N)*

*Edited on ***January 17, 2015, 9:19 pm**