This question is inspired by my inability to make any progress on Will Jagy's question. 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.

This question is inspired by my inability to make any progress on Will Jagy's question. 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.