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Prime and Consecutive Numbers


Date: 11/16/2001 at 11:01:27
From: Debbie Price
Subject: Prime factors

Why is it that 3, 5, and 7 are the only numbers that appear to be 
prime and consecutive?


Date: 11/16/2001 at 11:37:06
From: Doctor Paul
Subject: Re: Prime factors

Because any other group of three consecutive odd numbers will contain 
a multiple of three, and will hence not be prime.

Suppose x, x+2, and x+4 are all odd, and x > 3. If x is a multiple of 
three, then we've shown that one of them isn't prime. So assume x is 
not a multiple of three.

Then x is either one more than a multiple of three or two more than a 
multiple of three.

If x is one more than a multiple of three, then x+2 is a multiple of 
three and is hence not prime.

If x is two more than a multiple of three, then x+4 is a multiple of 
three and is hence not prime.

I hope this is clear.  Please write back if you'd like to talk about 
this some more.

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


Date: 11/16/2001 at 11:37:51
From: Doctor Jubal
Subject: Re: Prime factors

Hi Debbie,

Thanks for writing Dr. Math.

If you have a series of numbers n, n+2, n+4, exactly one of them must 
be divisible by three. Since every third number is divisible by three, 
any number can be written in one of these forms: 3k, 3k+1, 3k+2.

If n is of the form 3k, then it is divisible by three. If n is of the 
form 3k+1, then n+2 = 3k+3, which is divisible by three. If n is of 
the form 3k+2, then n+4 = 3k+6, which is divisible by three.

3 is the only prime divisible by three, so 3, 5, 7 is the only series 
n, n+2, n+4 where all three numbers are prime.

Does this help?  Write back if you'd like to talk about this some
more, or if you have any other questions.

- Doctor Jubal, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory
Middle School Prime Numbers

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