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• ## knowing when both 6k -1 and 6k + 1 are prime

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• Message: 2 Date: Sun Jun 4, 2006 10:31 am (PDT) From: develator81 develator81@gmail.com Subject: Trying to find when 6k+1 returns a prime I started with this
Message 1 of 1 , Jun 4, 2006
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Message: 2
Date: Sun Jun 4, 2006 10:31 am (PDT)
From: "develator81" develator81@...
Subject: Trying to find when 6k+1 returns a prime

I started with this question: It is possible to find a number "k"
that: 6k+1 returns a prime, in a way that we are sure of it
primality, with no need of testing it?
I'm speaking of "k" that are non zero positives integers.

I studied the numbers of the form: 6k+1 and 6k+1
I separated these numbers into three categories:
a) 6k+1 or 6k-1 that are primes
B) 6k+1 that aren't primes
c) 6k-1 that aren't primes.
In the categories "B" and "c", I wonder why those "k" don't return a
prime. Is there any rule for them?

Yes.

You have apparently derived one simple rule.

Here is another.

Suppose 6 k - 1 is composite.

Then for some r1 and s1 both > 1,

6 k - 1 =

Write r1 = 6 r2 + r3
s1 = 6 s2 + s3

r3 s3 = -1 = 5 mod 6.

Possible values of r3 and s3

r3 s3
1 5
5 1

Because of the symmetry, we can write

r1 = 6 r2 -1
s1 = 6 s2 + 1

r1 s1 = (6 r2 -1) (6 s2 + 1) = 36 r2 s2 + 6 r2 - 6 s2 - 1 = 6 ( 6 r2 s2 + r2
- s2) - 1

So if k is NOT a number of either of the forms

6 r2 s2 + r2 - s2,

6 r2 s2 - r2 _ s2

for both r2 and s2 > 0,

then 6 k - 1 is prime.

Likewise,

if k is NOT a number of either of the forms

6 r2 s2 - r2 - s2,

6 r2 s2 + r2 + s2,

Then 6 k + 1 is prime.

Combining these,

If k is NOT a number of any of the four forms,

6 r2 s2 - r2 - s2,

6 r2 s2 - r2 + s2,

6 r2 s2 + r2 - s2,

6 r2 s2 + r2 + s2

then

6 k -1 and 6 k + 1 are twin primes.

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