Re: [PrimeNumbers] 271, 577, 661, ...

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• ... The k-tuple conjecture predicts infinitely many cases of your form. The form looks a bit arbitrary. I doubt it has been investigated before. Most prime
Message 1 of 2 , May 11, 2011
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julienbenney wrote:
> The sequence corresponds to numbers k for which 15k-14, 15k-8, 15k+8
> and 15k+14 are all prime but 15k-4, 15k-2, 15k+2 and 15k+4 are all
> composite.

> I know that it is hard with my lack of money to afford software for me to
> check for further numbers in this sequence, so I wonder if anybody has
> noticed such patterns before? Are there an infinite number of such numbers?

The k-tuple conjecture predicts infinitely many cases of your form.
The form looks a bit arbitrary. I doubt it has been investigated before.

Most prime search software is free or written by the searchers.
This took less than a second with a simple script in the free PARI/GP:

{for(k=0,100000,
if(isprime(15*k-14)&isprime(15*k-8)&isprime(15*k+8)&isprime(15*k+14)&
!isprime(15*k-4)&!isprime(15*k-2)&!isprime(15*k+2)&!isprime(15*k+4),
print1(k", ")))}

271, 577, 661, 725, 831, 907, 2195, 2579, 3195, 3279, 4681, 4939,
5169, 5357, 6661, 7409, 7639, 9713, 12037, 14087, 14597, 15495,
15991, 16159, 16305, 16455, 17365, 17509, 17589, 18601, 18981,
19833, 20071, 20669, 20725, 21163, 22857, 23075, 24937, 25651,
25849, 25883, 26115, 26301, 27625, 31433, 31461, 31917, 32471,
32869, 33379, 33847, 34543, 35123, 36165, 36895, 38231, 38371,
42313, 42563, 42599, 43433, 43691, 44083, 47633, 49437, 49453,
50493, 50859, 51371, 52125, 53015, 53399, 55731, 57075, 59243,
59265, 60237, 60505, 61359, 63375, 64285, 65621, 66357, 66393,
69289, 69345, 70395, 70681, 71915, 72319, 72693, 73565, 73881,
77535, 78327, 79861, 80069, 81429, 82079, 82353, 82771, 83645,
84991, 86089, 88239, 88259, 89045, 89231, 91647, 91733, 92339,
93299, 94749, 94769, 95847, 96273, 97827, 99377,

More efficient software could easily find millions of cases.
k = 2253716*599#/5+67 gives a case with 252-digit primes.

--
Jens Kruse Andersen
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