```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 thatif 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 allx in S,   x+C  is a prime.With C = 389,232,355,162,471from your computations, all  the numbers    x+C  forx 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 totwin primes.The set S has a maximum difference between elements of360, and has a cardinality of 12.I'm wondering how large in cardinalitya "translated set of primes" candidate set T can beif, say, the maximum difference between elements of T isat 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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