Interesting prime curio - 111119
- What is special about the prime 111119?
It is the smallest prime which does not divide some
10-digit pandigital number. (A pandigital number contains
at least one of each of the 10 digits, and by convention,
does not start with a '0'.) There are 9*9! (3265920)
10-digit pandigital numbers, and not one of them is
divisible by 111119. If their residues modulo 111119
were equally distributed, the "chance" of this happening
would be (111118/111119)^3265920, or about 1 in 5.8*10^12.
Considering that 111119 is the 10545th prime, it's clear
there is more than random luck at work here.
As an exercise for the reader... show that every prime has a
perfect power which is pandigital. Note that you don't need
to give a constructive method, although it's certainly
possible to do so.
Hint: show that log(p)/log(10) is irrational for any prime p.