General Observation on Prime NumbersDate: 09/03/2004 at 19:31:48 From: Stanley Subject: prime numbers Is it true that all prime numbers greater than 5 are of the form 6n + 1 or 6n - 1? I read this on a website, but it's hard to believe. Date: 09/05/2004 at 15:02:28 From: Doctor Beryllium Subject: Re: prime numbers Hi Stanley, Yes, it is the case that all primes greater than 6 are either of the form 6k + 1 or 6k - 1. Consider that we can divide the set of integers into the following classes: class 0 6k - 1 class 1 6k class 2 6k + 1 class 3 6k + 2 class 4 6k + 3 class 5 6k + 4 class 6 6k + 5 Further notice that the the difference between integers of class 0 and class 6 is 6 (this is because -1 is congruent to 5 modulo 6). That means that this cycle of integer classes will repeat over and over as we count through the integers with increasing values of k. Now integers of class 1 have 6 as a factor. Integers of class 3 have 2 as a factor. Integers of class 4 have 3 as a factor. Integers of class 5 have 2 as a factor. This leaves us with the fact that all primes must either be from class 2 or class 0 or class 6. But we know integers from class 6 are the same integers as from class 0. This shows that all primes must be of one of the following forms: 6k - 1 == 6k + 5 (congruent modulo 6) or 6k + 1 - Doctor Beryllium, The Math Forum http://mathforum.org/dr.math/ |
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