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

Charles Greathouse charles.greathouse at case.edu
Thu Dec 17 19:02:21 CET 2009

```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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```