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Re: primes in the arithmetic sequence 1, 31, 61, 91, 121, ... 30k+1 ...
Posted:
Apr 24, 2013 9:41 AM


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.



