This question is inspired by my inability to make any progress on [Will Jagy's question](https://mathoverflow.net/questions/23943). 
Giving a positive answer to this question should be strictly easier than proving Jagy's conjectures.


Suppose that $K/\mathbb{Q}$ is an imaginary quadratic extension. Let $\chi$ be the corresponding quadratic character. Suppose that there exist $k$ consecutive integers such that $\chi(a)=\chi(a+1)=\ldots=\chi(a+k-1)=1$. Do there necessarily exist infinitely many integers $b$ such that $b$, $b+1$, ... and $b+k-1$ are all norms of ideals in $\mathcal{O}\_K$?


For example, the first interesting case is to determine whether there are infinitely many $b$ such that, in the prime factorizations of both $b$ and $b+1$, those primes which are $3$, $5$ or $6$ modulo $7$ all occur an even number of times. 


The motivation here is that Jagy's questions seem to mix a "sieve" question and a "class group" question. My question aims to isolate the sieve problem as its own challenge.



Achieving consecutive integers as norms from a quadratic field
The answer to Speyer's question as stated is no, this need not be 
the case. To see this let $p\equiv 3\pmod 4$ be a prime and consider 
the associated imaginary quadratic field ${\Bbb Q}(\sqrt{-p})$. Note that 
the associated quadratic character is simply the Legendre symbol $(\frac{n}{p})$. 


Then as observed by David Hansen, if $p$ is sufficiently large, there will 
certainly exist $k$ consecutive quadratic residues $\pmod p$ so that the 
hypothesis in Speyer's question is satisfied. But on the other hand if 
say $3$ is a non-residue $\pmod p$ then among any six integers there would 
be an integer divisible by $3$ and not $9$ and this integer cannot be 
the norm of an ideal. 


The right conjecture is that if the primes below $k$ are split in 
the number field $K$ then there must be infinitely many strings of $k$ 
consecutive numbers all of which are norms of ideals in $K$. This does not 
seem easy to prove, and I think (for $k\ge 3$) is comparable in strength 
to the Hardy-Littlewood prime $k$-tuples conjecture. One can also deduce this 
result from the (generalized) Hardy-Littlewood conjectures: If $D$ is 
such a discriminant, consider the $k$-tuple $|D|k! n +1, \ldots, |D|k! n+k$. 
These will all satisfy $\chi(|D|k! n+\ell) =1$ (since $\chi(\ell)=1$ for all $\ell \le k$ by assumption), and it should be possible (by Hardy-Littlewood) to arrange all 
the $|D|k!n/\ell +1$ to be primes. That does the job. 


Alternatively, one can argue as in Hardy-Littlewood and write down conjectures 
for the number of consecutive $k$-tuples that are norms of ideals. One would also 
guess that these could be made to lie in arbitrary classes of the class group, 
and that would answer the motivating question of Jagy on quadratic forms. 


