[seqfan] Re: On a claim by Chun-Xuan Jiang.
Charles Greathouse
charles.greathouse at case.edu
Thu Mar 15 07:22:27 CET 2012
Under Dickson's conjecture (which is widely believed and subsumed
under a number of more recent conjectures, including that of Bateman,
Horn, & Stemmler), there are infinitely many such primes. The sequence begins
1, 121, 380, 506, 511, 3796, 5875, 6006, 8976, 9025, 9186, ...
and does not appear in the OEIS.
This GP code (gp is freely available at http://pari.math.u-bordeaux.fr/ )
for(k=1,1e7,if(isprime(6*k+1)&&isprime(12*k+1)&&isprime(18*k+1)&&isprime(36*k+1)&&isprime(72*k+1),print1(k",
")))
will produce the first 770 terms. (Replace 1e7 with a larger number
if you'd like more.)
Charles Greathouse
Analyst/Programmer
Case Western Reserve University
On Thu, Mar 15, 2012 at 1:30 AM, Ed Jeffery <lejeffery7 at gmail.com> wrote:
> Sorry to post again so soon.
>
> In his paper "Disproof of Reimann's Hypothesis," which is supposed to have
> appeared in Algebras, Groups and Geometries, Vol 21, 2004, see
> http://vixra.org/pdf/1004.0028v1.pdf (page 10), Chun-Xuan Jiang claims that
> there exist infinitely many integers k such that
>
> p1 = 6*k + 1,
> p2 = 12*k + 1,
> p3 = 18*k + 1,
> p4 = 36*k + 1,
> p5 = 72*k + 1,
>
> and p1, p2, p3, p4, p5 are all primes. The numbers n = p1*p2*p3*p4*p5 are a
> class of so-called Carmichael numbers.
>
> I tried to calculate the sequence of such k manually but failed, since I
> don't have anything like Mathematica. It is obviously true for k = 1, but I
> got tired of checking for primality at around k = 85. So I wonder:
>
> What are the next several values of k?
>
> Is the sequence indeed infinite?
>
> Is the sequence in the OEIS database?
>
> Regards.
>
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