Twin Prime Numbers

Date: 3/11/96 at 15:19:1
From: KHU
Subject: Twin prime numbers

You know that a prime number is a whole number greater than 
1 whose only whole number divisors are 1 and itself.  You may not 
know that there are also such things as twin prime numbers.  These 
are pairs of prime numbers that are only 2 apart, such as 5 and 7, or 
17 and 19, or 41 and 43.  There are many interesting things to notice 
about twin primes.  

This problem mostly concerns the following observation about twin 
primes.

Start with any pair of twin primes, except the combination 3 and 5.  
If you multiply the two primes together and add 1 to the product, 
you always get a number which is a perfect square and a multiple 
of 36.

1. Experiment with some other pairs of twin primes, as well as with 
pairs of numbers that are not twin primes, and try to get some 
insight into what is happening.

2. Prove the two facts about the process of multiplying twin primes 
and adding 1.  You will need to use some variables in your proof!

3. See if you can figure out anything else interesting about twin 
primes.

Date: 5/16/96 at 20:48:23
From: Doctor Ken
Subject: Re: Twin prime numbers

Hello!

Here's one thing to try: take ANY two numbers that differ by 2.  
Now multiply them together and add 1.  What do you notice about 
what you get?  Can you prove it?  Hint: call the two numbers (n-1) 
and (n+1).

Now, about the divisible by 36 thing.  You know that the original 
numbers (n-1) and (n+1) are prime, right?  In particular, that means 
that neither 2 nor 3 divides either one of them.  So what can you say 
about whether 2 or 3 divides n?  And what does that tell you about 
(n-1) times (n+1) plus 1?

-Doctor Ken,  The Math Forum