nt.number theory - Achieving consecutive integers as norms from a quadratic field

Something small, but maybe useful, which no one seems to have pointed out: as $ptoinfty$, $(mathbb{Z}/pmathbb{Z})^{times}$ contains arbitrarily long strings of consecutive quadratic residues. Indeed, the function

$b(a)=2^{-k}(1+(frac{a}{p}))(1+(frac{a+1}{p}))dots(1+(frac{a+k-1}{p}))$

is $1$ or $0$ according to whether $(a,a+1,dots,a+k-1)$ is a $k$-term string of quadratic residues or not. Summing over $(mathbb{Z}/pmathbb{Z})^{times}$, expanding out and using the bound of Weil,

$lvert sum_{a in (mathbb{Z}/pmathbb{Z})^{times}} (frac{(a+b_1)(a+b_2)dots (a+b_r)}{p}) rvert leq 2rsqrt{p},$

which holds if at least one $b_i$ is distinct from all the others, we derive

$sum_{a in (mathbb{Z}/pmathbb{Z})^{times}}b(a)=2^{-k}p+O(ksqrt{p})$.

The error term here comes from the fact that when we expand out $b(a)$ and sum, we'll get the obvious main term, plus $2^{-k}$ times a sum of $2^{k}-1$ Weil sums, each of which is bounded by $2ksqrt{p}$.

Anyway, the main term dominates the error term if $k2^{k}=o(sqrt{p})$, which certainly holds if (say) $k=o(log{p}).$