Number Sequence Problem

Date: 08/11/97 at 14:58:54
From: Jon Mann
Subject: Number sequence problem

I have a number sequence but can not find out the pattern. Here are 
the numbers:

7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 
193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 
433, 461, 487, 491, 499, 503, 509, 541, 571, 577, ETC....

Dr. Math, you would be my hero if you got this sequence!

J. M. 

Date: 08/15/97 at 15:04:54
From: Doctor Rob
Subject: Re: Number sequence problem

Dear J. M.,

This is an interesting question!

Your sequence is the set of primes p for which 10 is a primitive root.
Equivalently, this is the set of primes p for which p does not divide
10^k-1 for any k with 0 < k < p-1.  Equivalently, this is the set of
primes p for which 1/p has repeating decimal period of length p-1.

 1/7 = .142857 142857 142857 ...     period 6,
1/17 = .0588235294117647 0588235294117647 ...   period 16,
1/19 = .052631578947368421 052631578947368421 ...   period 18, etc.

The length of the period must divide p-1, but is not always equal:

 1/3 = .3 3 3 3 ... period 1 | 2 = 3-1,
1/11 = .09 09 09 ... period 2 | 10 = 11-1,
1/13 = .076923 076923 ... period 6 | 12 = 13-1,
1/31 = .032258064516129 032258064516129 ... period 15 | 30 = 31-1, 
etc.

You can look up other sequences to find their rules by consulting the
following URL:

   http://www.research.att.com/~njas/sequences/index.html   

There you can access a huge catalog of sequences online.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/