"Accidental Order" in Pi, e

Date: 04/12/99 at 20:44:06
From: collins titus Ngeno
Subject: Prime numbers!

The occurence of the sequence 1234567890 in the decimal expansion for 
pi or e may be called "accidental order."
 a) Show that this sequence appears in the digital interior of 
infinitely many prime numbers.
 b) Does the sequence consisting of the first billion digits of pi 
appear in the digital interior of infinitely many prime numbers?

Please help.
Titus

Date: 04/14/99 at 13:02:43
From: Doctor Nick
Subject: Re: Prime numbers!

Hi Titus,

I have to assume you are studying number theory, and that you know of 
Dirichlet's result on primes in arithmetic progression. This states 
that if r and s are relatively prime, then the set of positive 
integers r, r+s, r+2*s, r+3*s, ... contains infinitely many primes.

We can use this result to answer questions like your a) and b). I'll 
do a simpler problem, and you can work out yours.

Let's show that there are infinitely many primes that have "12" in
their digits. We notice that primes can't end in "12," but they can 
end in "121." 

So, consider the arithmetic progression with r=121, s=1000:

   121, 1121, 2121, 3121, 4121, ..., 10121, 11121, ... .
   
Since 121 and 1000 are relatively prime, this progression contains
infinitely many primes; hence, there are infinitely many primes which 
contain "12" in their digits.

Notice that this doesn't come anywhere near accounting for _all_ 
primes that have "12" in their digits, but that's not needed. Also 
notice that we could have used r=123, s=1000, or r=1201, s=10000, etc.

Now, you just have to apply this method to the larger digital 
sequences of a) and b). Write back if you get stuck.

Have fun,

- Doctor Nick, The Math Forum
  http://mathforum.org/dr.math/