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### p, p+8, p+22 Not Prime

```
Date: 10/16/2001 at 07:37:17
From: Scott S
Subject: Proof On Prime Numbers

Could you help me with this question?:

Prove that there is no positive integer p such that each of the
numbers:

p
p+8
p+22

is prime.

Any help would be much appreciated.
Cheers,
Scott
```

```
Date: 10/16/2001 at 08:38:49
From: Doctor Paul
Subject: Re: Proof On Prime Numbers

If p is not prime, we're done. So assume p is prime. Then p is either
one more than a multiple of three, or p is two more than a multiple of
three.

In other notation:

p = 1 mod 3 or p = 2 mod 3.

If p = 1 mod 3 then p+8 = 0 mod 3 and is therefore divisible by
three and is hence not prime.

If p = 2 mod 3 then p+22 = 0 mod 3 and is therefore divisible by
three and is hence not prime.

- Doctor Paul, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Number Theory

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