# [seqfan] Re: Primes of the form kn+1

Jonathan Post jvospost3 at gmail.com
Thu Dec 17 19:08:56 CET 2009

```Xylouris, updating Heath-Brown's result on Linnik's constant, proved
in 2009 that prime(k) << k^5.2.

On Thu, Dec 17, 2009 at 10:02 AM, Charles Greathouse
<charles.greathouse at case.edu> wrote:
> Linnik (1944) showed that there exist k and L such that the least
> prime solution to
> is at most k * d^L (for a, b relatively prime).  Heath-Brown [1] shows
> that L can be taken to be 5.5.  You're asking if we can take L = 2 and
> k = 1.  L = 1 + eps is known under the GRH [1], so it seems likely.
>
> 1. http://eprints.maths.ox.ac.uk/166/
>
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
>
> On Thu, Dec 17, 2009 at 12:42 PM, Michael Porter
> <ic_designer at verizon.net> wrote:
>> Is it true that for every positive integer n, there is a prime of the form kn+1 with 1<=k<=n?
>>
>> I submitted the following PARI program for sequence A089727:
>> A089727(n) = {local(k);k=n;while(!isprime(n*k+1),k--);n*k+1}
>> and then I started wondering if the while loop would always terminate.
>>
>> Thanks,
>> Michael
>>
>>
>>
>>
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>
>
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```