- Hi,

If you take the gaps between consecutive odd primes modulo 6, they are either 0, 2 or 4. For 4 consecutive primes, out of 27 permutations, only 15 appear. Can someone point me to an elementary proof of why this is so? Some of the missing ones are obvious (2,2,2). I am particularly interested in (2,0,2) and (2,4,2). The first occurs and the second does not.

Many thanks,

David - --- On Fri, 9/2/11, pbtoau <PbtoAu@...> wrote:
> If you take the gaps between consecutive odd primes modulo

I'm dismayed to hear of the unfortunate removal of the set {5,7,11,13} from the list of primes, they will be sorely missed. I couldn't give a monkey's about losing {101,103,107,109} though.

> 6, they are either 0, 2 or 4. For 4 consecutive

> primes, out of 27 permutations, only 15 appear. Can

> someone point me to an elementary proof of why this is

> so? Some of the missing ones are obvious

> (2,2,2). I am particularly interested in (2,0,2) and

> (2,4,2). The first occurs and the second does not.

Phil - --- In primenumbers@yahoogroups.com,

"pbtoau" <PbtoAu@...> wrote:

> I am particularly interested in (2,0,2) and (2,4,2).

[5,7,11,13] has gaps [2,4,2] in Z and hence in Z/Z6.

> The first occurs and the second does not.

David - pbtoau wrote:
> If you take the gaps between consecutive odd primes modulo 6, they are

I think you swapped the two tuples.

> either 0, 2 or 4. For 4 consecutive primes, out of 27 permutations,

> only 15 appear. Can someone point me to an elementary proof of why this

> is so? Some of the missing ones are obvious (2,2,2). I am particularly

> interested in (2,0,2) and (2,4,2). The first occurs and the second does

> not.

As others have pointed out, the second (2,4,2) occurs.

But the first (2,0,2) does not occur.

The 12 which do not occur:

(0, 2, 2), (0, 4, 4), (2, 0, 2), (2, 2, 0), (2, 2, 2), (2, 2, 4),

(2, 4, 4), (4, 0, 4), (4, 2, 2), (4, 4, 0), (4, 4, 2), (4, 4, 4).

They are all inadmissible modulo 3, meaning that 3 will divide at least one

of four numbers with those gaps.

It may be easier to see what happens if you for each of the four numbers

write the difference from the first number modulo 6.

The difference of the first number from itself is 0.

Gaps (2, 0, 2) corresponds to these differences to the first number modulo 6:

[0, 2, 2+0, 2+0+2] = [0, 2, 2, 4]

This as inadmissible modulo 3 because all values modulo 3 occur.

See more at http://primes.utm.edu/glossary/xpage/ktuple.html

You can check whether a tuple written on this form is admissible here:

http://primes.utm.edu/glossary/includes/ktuple.php

For [0, 2, 2, 4] it says: "fails at prime 3 (term 4)"

This means the tuple becomes inadmissible modulo 3 when you get to the

4th term.

--

Jens Kruse Andersen