Conjecture

[Dean Hickerson]

Artur <grafix at csl.pl> wrote:

> Who is able to proove conjecture from
> A13464  <http://www.research.att.com/%7Enjas/sequences/A134641> ?

The conjecture is that if a number's base b expansion is a permutation
of  {0,1,2, ..., b-1},  and the number is prime, then  b<=3.  The proof is
easy:  Every number is congruent (mod b-1) to the sum of its digits in
base b.  So if the digits of n are a permutation of {0,1,2, ..., b-1},
then n is congruent to  b(b-1)/2 (mod b-1).  Hence n is divisible by
gcd(b(b-1)/2, b-1), which equals  b-1  if b is even,  (b-1)/2  if b is odd.
If  b>3,  then this gcd is larger than 1 and less than n, so n isn't prime.

Dean Hickerson
dean at math.ucdavis.edu