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Dom
Dom
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Let $p > 3$ be prime. Is is true that there exists $x \in \mathbb{Z}_p$ such that $$ (1+x^2)^3-1 $$ is not a square in $\mathbb{Z}_p$? EquivalentlyIn particular, when $-1$ is not a square in $\mathbb{Z}_p$, can we show that the equation $$ -y^2 = (1+x^2)^3-1 $$ always has non-trivial solutions $(x,y) \ne (0,0)$ in $\mathbb{Z}_p$?

Let $p > 3$ be prime. Is is true that there exists $x \in \mathbb{Z}_p$ such that $$ (1+x^2)^3-1 $$ is not a square in $\mathbb{Z}_p$? Equivalently, can we show that the equation $$ -y^2 = (1+x^2)^3-1 $$ always has non-trivial solutions $(x,y) \ne (0,0)$ in $\mathbb{Z}_p$?

Let $p > 3$ be prime. Is is true that there exists $x \in \mathbb{Z}_p$ such that $$ (1+x^2)^3-1 $$ is not a square in $\mathbb{Z}_p$? In particular, when $-1$ is not a square in $\mathbb{Z}_p$, can we show that the equation $$ -y^2 = (1+x^2)^3-1 $$ always has non-trivial solutions $(x,y) \ne (0,0)$ in $\mathbb{Z}_p$?

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Dom
Dom
  • 43
  • 5

Polynomial that is not always a square over $\mathbb{Z}_p$

Let $p > 3$ be prime. Is is true that there exists $x \in \mathbb{Z}_p$ such that $$ (1+x^2)^3-1 $$ is not a square in $\mathbb{Z}_p$? Equivalently, can we show that the equation $$ -y^2 = (1+x^2)^3-1 $$ always has non-trivial solutions $(x,y) \ne (0,0)$ in $\mathbb{Z}_p$?