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### "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?

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/
```
Associated Topics:
High School Number Theory
High School Transcendental Numbers

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