consecutive integers that are not the sum of 2 squares.

Question

Are there any $n\in\mathbb{N}$ such that no element $k\in\{n,n+1,n+2,...,n+2017\}$ can be expressed as $a^2+b^2$ for some $a,b\in\mathbb{Z}$?

Answer 1

Yes. Let $p_1, p_2, \cdots, p_{2018}$ be distinct primes of the form $4k+3$. By Chinese Remainder Theorem there exists $n$ such that

$$ \begin{aligned} n &\equiv p_1 &\pmod{p_1^2}\\ n &\equiv p_2 - 1 &\pmod{p_2^2}\\ &\vdots\\ n &\equiv p_{2018} - 2017 &\pmod{p_{2018}^2}. \end{aligned} $$

Then $\{n, n+1 , \cdots, n+2017\}$ satisfies your requirements.

Answer 2

The set $E$ of numbers of the form $a^2+b^2$ has density zero, in particular there are at most $\frac{CN}{\sqrt{\log N}}$ numbers of such form in $[1,N]$ for large values of $N$. Assuming that there is a number of such form in each interval with length $2018$ gives that the density of $E$ is at least $\frac{1}{2018}$, contradiction.