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Topic: primes in the arithmetic sequence 1, 31, 61, 91, 121, ... 30k+1 ...
Replies: 4   Last Post: Apr 24, 2013 11:24 AM

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David Bernier

Posts: 3,240
Registered: 12/13/04
Re: primes in the arithmetic sequence 1, 31, 61, 91, 121, ... 30k+1
...

Posted: Apr 24, 2013 9:41 AM
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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.



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