Copyright ©1997 by Paul Niquette. All rights reserved. 
The puzzle
calls for only five
cases of odd decimal numbers with a common cipher...
111...1
Each is divisible by its respective common cipher, which, in the case of 111...1, is 1. The sophisticated solver knows that the divisor 1 cannot discriminate between prime and nonprime integers. The divisor 11 can, though. A dividend 111...1 will be
divisible by
11 whenever
the number of digits is even  producing a quotient
of 1010...01.
That will be true for half of the odd decimal numbers
with the common cipher
1.


Paradox in Numbers
Still, 90% of them are nonprimes. Huh?The sophisticated solver who cares will readily see that the outcome is the same for the all interpretations of the word 'number' substituted for the word 'integer.'
More than five years after the
publication of Prime
Numbers are Odd, the following email
message was received from
John Swanson, sophisticated puzzle solver and world
renown bridge champion:
Swanson's "at least 90%" is quite appropriate. That really bums me out. Meanwhile, an old engineer has speculated elsewhere that no more than about 12% (1 in 8.3) of all integers are primes. If that proportion holds for primes larger than 9,883, then about 88% of all integers are nonprimes, and integers of the form 111...1 are quite atypical. And then, on the occasion of his 70^{th}
birthday
in December of 2003, that same engineer read the
following headline
"Student Finds Largest Known Prime Number," which
inspired yet another
puzzle.
