Twin Prime NumbersDate: 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 |
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