Copyright ©1997 by Paul Niquette. All rights reserved.
|The puzzle calls for only five
cases of odd decimal numbers with a common cipher...
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 non-prime integers. The divisor 11 can, though.
A dividend 111...1 will be divisible by
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
Paradox in Numbers
Still, 90% of them are non-primes. 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 e-mail 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 non-primes, and integers of the form 111...1 are quite atypical.
And then, on the occasion of his 70th birthday
in December of 2003, that same engineer read the following headline
"Student Finds Largest Known Prime Number," which inspired yet another