Inspired by this recent question, I wondered if a similar result is true for quadratic non-residues, namely, if it is true that for every $k \in \mathbb{N}$ there exists a prime $p$ such that exists $k$ consecutive quadratic non-residues mod $p$? or even a stronger result like for every $k \in \mathbb{N}$ there exists $N$ such that for every prime $p > N$ we have $k$ consecutive quadratic non-residues mod $p$?

Obviously, the idea used in the link could not be adapted to this case because it relies on the fact that the product of two quadratic residues is a quadratic residue, which is not true for quadratic non-residues. I searched the web and did not find anything relevant on this question. Does anybody have any ideas?

Thank you!

PS: If anyone could give me a reference to a book that approaches this kind of problems (consecutive quadratic or non-quadratic residues), I would be grateful.

Inspired by this recent question, I wondered if a similar result is true for quadratic non-residues, namely, if it is true that for every $k \in \mathbb{N}$ there exists a prime $p$ such that exists $k$ consecutive quadratic non-residues mod $p$? or even a stronger result like for every $k \in \mathbb{N}$ there exists $N$ such that for every prime $p > N$ we have $k$ consecutive quadratic non-residues mod $p$?

Obviously, the idea used in the link could not be adapted to this case because it relies on the fact that the product of two quadratic residues is a quadratic residue, which is not true for quadratic non-residues. I searched the web and did not find anything relevant on this question. Does anybody have any ideas?

Thank you!

PS: If anyone could give me a reference to a book that approaches this kind of problems (consecutive quadratic or non-quadratic residues), I would be grateful.