Prime Numbers Are Odd

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
333...3
555...5
777...7
999...9

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 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.
 
    One-half of one-fifth equals 10%, leaving 90% as non-prime. QED

 


Paradox in Numbers
hat percentage of all integers have common decimal ciphers?
  • There are 90 two-digit integers (10, 11, 12, ... 99); nine of them have common ciphers: (11, 22, ... 99), which amounts to 10 %.
  • There are 900 three-digit integers (100, 101, 102, ... 999); nine of them have common ciphers (111, 222, ... 999), which amounts to 1 %.
  • There are 9,000 four-digit integers (1000, 1001, 1002, ... 9999); nine of them have common ciphers (1111, 2222, ... 9999), which amounts to 0.1 %.
  • There are 9 * 10n-1 n-digit integers; nine of them have common ciphers, which amounts to 10(2 - n) %.
Tempting, isn't it, to just split an infinitive -- er, to just add up the percentages:
    10 + 1 + 0.1 + 0.01 ... = 11.111... % ~~~~ BLAGH!
The correct answer we seek is in the form...
    100 (9 + 9 + 9 + ... ) / (90 + 900 + 9,000 + ...) %
...which amounts to exactly 0% with any desired degree of accuracy.
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.'
 
For more observations about primes, see 
Prime Number Numbers.

Epilog

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:
 

Paul,

In the "Prime Numbers are Odd" puzzle, I believe that the problem is not to show that "90% of all odd decimal numbers of the form ddd...d are not prime" but to show that at least 90% of those numbers are non-prime. 

Now, the number of digits in a number of the form 111...1 [let's call this 1(n)] must be prime for the number to have a possibility of being prime.  Else the number 1(m) where m is a factor of n will be a divisor of 1(n).

Best
John

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 puzzle.
 



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