Take $S=\{x:5\nmid x\}$. Among every 7 consecutive numbers, at most two of them are divisible by $5$, so $S$ is dense. Now, suppose there exists $a,b,c\in S$ such that $a^2+b^2=c^2$. As $a^2,b^2,c^2\in\{1,4\}\pmod{5}$, this is clearly impossible.