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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 $a,b\in\mathbb{Z}$?

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closed as off-topic by man and laptop, Namaste, The Phenotype, Xam, Shailesh Feb 3 '18 at 5:26

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  • $\begingroup$ The set of members of $\Bbb N$ that are sums of 2 non-zero squares of integers has asymptotic density $0$ in $\Bbb N,$ so we can answer that there are infinitely many such $n$ without giving a single example.( Not the most direct method!) $\endgroup$ – DanielWainfleet Feb 2 '18 at 22:24
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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.

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  • $\begingroup$ Note that this will generate a number with several thousand digits. The smallest n that satisifies the condition is probably much smaller than one generated by this method. Compare, for example, the proof that there are arbitrarily large prime gaps because none of $k! + 2, k! + 3,\ldots, k! + k$ are prime. This isn't meant to say that the answer is wrong, just to point out a natural further question. $\endgroup$ – Michael Lugo Feb 2 '18 at 18:09
  • $\begingroup$ You're right. It might be another interesting question to ask what is the magnitude of the smallest solution. $\endgroup$ – Hw Chu Feb 2 '18 at 18:12
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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.

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  • $\begingroup$ My comment to the Q was this, but I made the comment before reading your A. $\endgroup$ – DanielWainfleet Feb 2 '18 at 22:26

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