Date: Apr 24, 2013 9:41 AM
Author: David Bernier
Subject: Re: primes in the arithmetic sequence 1, 31, 61, 91, 121, ... 30k+1<br> ...
On 04/22/2013 02:22 PM, christian.bau wrote:

>> I.e. Out of thirteen consecutive integers from the sequence

>> of the 30k+1, can we get at least 12 primes out

>> of the thirteen numbers, for the right choice

>> of the 13 consecutive numbers ?

>

> Of those 13 consecutive numbers, one or two are divisible by 7; one or

> two are divisible by 11, one is divisible by 13, at most one divisible

> by 17 etc. To have only one divisible by 7, it must be the middle one.

> To have only one number composite, that number must also be divisible

> by 11 and 13. 1001 = 7x11x13. So you need to check

>

> (1001 * (30k + 11)) - 180, -150, -120, -90, -60, -30, +30, +60,

> +90, +120, +150, +180.

>

> 389,232,355,162,471 + 0, 30, 60, 90, 120, 150, 210, 240, 270, 300,

> 330, 360 are all primes.

>

I was enthused that two people (Don Reble and yourself)

found examples of what I was looking for and posted.

If S = {0,30,60,90,120,150,210,240,270,300,330,360}

then if the prime p is set to p=2,

none of the numbers in S+1 is congruent to 0 (mod 2).

If p=3,

none of x in S+1 is congruent to 0 (mod 3).

If p=5

(same with S+1)

If p=7, none of the x in S+2 is congruent to 0 (mod 7).

If p=11, none of the x in S+5 is congruent to 0 (mod 11)

if p>11 is a prime, there exits n_p such that

if x is in S, then x + n_p == 0 (mod p)

(Note: if p> 361, this is not hard to see).

So, there's no modular arithmetic "obstruction"

to the existence of C>0 such that for all

x in S, x+C is a prime.

With C = 389,232,355,162,471

from your computations, all the numbers x+C for

x in S = {0,30,60,90,120,150,210,240,270,300,330,360}

are prime.

S could be called a "translated set of primes" candidate,

for instance.

In the same way, if S_2 = {0, 2}, S_2 is a

"translated set of primes" candidate: connected to

twin primes.

The set S has a maximum difference between elements of

360, and has a cardinality of 12.

I'm wondering how large in cardinality

a "translated set of primes" candidate set T can be

if, say, the maximum difference between elements of T is

at most 360.

It's a way of looking at potential primes clumpiness over "small"

distances.

David Bernier

--

Jesus is an Anarchist. -- J.R.