Skip to content

Proof: Primes are 6n +- 1

July 22, 2011

All primes (except 2 and 3) can be written in the form 6n plus or minus one

Proof
Let p be a prime number > 3
Let n be an appropriate integer.

Case 1
If p = 6n, then p would have a factor of 6 and therefore p could not be prime.

Case 2
If p = 6n + 2, then p would have a factor of 2 and therefore p could not be prime.

Case 3
If p = 6n + 3, then p would have a factor of 3 and therefore p could not be prime.

Case 4
If p = 6n – 2, then p would have a factor of 2 and therefore p could not be prime.

This only leaves p being able to be 6n + 1 or 6n – 1

It’s a proof by exhaustion as there are no other possible cases.

( NB. Converse is not true. 65 = 6 x 11 – 1 and 65 is not prime.)

This proof is taken from the Mathematical Association’s book “Problem Pages 14-19”.
[I’m now considering joining the MA.]

Advertisements

From → Uncategorized

Leave a Comment

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: