[seqfan] Re: Is this sequence duplicate of A088192 Distance between the primes and the largest quadratic residues modulo the primes

[Max Alekseyev]

A088192(n) can be defined defined as the smallest d>0 such that -d is
a quadratic residue modulo p = prime(n).
At the same time, in your sequence a(n) = d if and only if p is
representable in the form x^2+d*y^2.
While -d being a quadratic residue modulo p is necessary condition for
such a representation, it is not sufficient. Additionally it is
required that a certain polynomial, denoted f_d(x), has zeros modulo
p. For details, see http://math.rice.edu/~av15/Files/Gauss.pdf
That is, in general we have a(n) >= A088192(n) but I currently do a
reason why it should be a(n) = A088192(n). So my bet is that there
exists a counterexample for this equality (in this counterexample,
f_d(x) would have no zeros modulo p).

Regards,
Max

On Mon, Oct 29, 2012 at 12:00 PM, Georgi Guninski <guninski at guninski.com> wrote:
> Is this sequence duplicate of A088192 Distance between the primes and the largest quadratic residues modulo the primes
>
> A088192 Distance between the primes and the largest quadratic residues modulo the primes
>
> I am trying to compute a(n)=smallest d s.t. the n-th prime is
> composite in Q[sqrt(-d)].
>
> Using idealfactor() the pari script is:
>
> {ndi(d,p)=#idealfactor(bnfinit(x^2+d),p)~==1}
>  forprime(p=2,300,for(d=1,p,if(!ndi(d,p),print1(d,",");break) ))
> 2,1,3,2,1,1,2,5,1,3,1,1,2,5,1,2,1,2,7,1,3,2,1,1,1,3,2,1,1,3,2,1,2,1,3,1,2,5,1,2,1,7,1,1,3,2,3,2,1,1,7,1,2,1,5,1,3,1,1,2,1,
>
> 1. Is this a correct way to compute it? (Checking for being an integer
> norm gives very few differences)
> 2. Is this the same as A088192?
>
> Thanks.
>
>
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