Squares in $\mathbb{Z}_p$ that are not squares in $\mathbb{Z}$

This is the famous fake square problem (or, at least, I love to call it this way).

Assume that $n\in\mathbb{N}^+$ is not a square but it turns to be a quadratic residue $\!\!\pmod{p}$ for any prime $p$. Let $q_1,\ldots,q_k$ be the primes appearing in the factorization of $n$ with an odd exponent ($k\geq 1$ since $n$ is not an integer square) and let $\eta_k$ be the least quadratic non-residue $\!\!\pmod{q_k}$. By Dirichlet's theorem there is a prime $r$ such that $$r\equiv 1\!\!\!\!\pmod{4},\;r\equiv 1\!\!\!\!\pmod{q_1},\;\ldots\;, r\equiv 1\!\!\!\!\pmod{q_{k-1}},\; r\equiv \eta_k\!\!\!\!\pmod{q_k}.$$ For such prime $r$, by the multiplicativity of the Legendre symbol and quadratic reciprocity we have:

$$\left(\frac{n}{r}\right)=\prod_{h=1}^{k}\left(\frac{q_h}{r}\right)=\prod_{h=1}^{k}\left(\frac{r}{q_h}\right)=-1$$ hence $n$ is not a quadratic residue $\!\!\pmod{r}$, contradiction.

One might wonder if there are integers that differ from any sum of two non-negative integer cubes, but can be represented as $x^3+y^3\pmod{p}$ for any prime $p\equiv 1\pmod{3}$. Since the non-trivial integer solutions of $a^3+b^3=c^3+d^3$ are known as the taxicab numbers, we may call the problem presented after the breakpoint as the fake taxi problem. (;P)