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Let $L$ be a cyclotomic field, and $P$ a prime ideal of $\mathcal{O}_L$.

1. is there any symbol for the equation $ax^2 + by^2 =1 \mod P$ and if so, is it computable in polynomial time? if $a$ is invertible modulo $P$ this is equivalent to $(1-by^2)/a$ is a square in $\mathcal{O}_L/P$.

2. is there any algorithm to find such $x$ and $y$ ? all such possible $(x,y)$?

3. an interesting case is: $a=b$ and $a$ is invertible modulo $P$.

Let $L$ be a cyclotomic field, and $P$ a prime ideal of $\mathcal{O}_L$.

1. is there any symbol for the equation $ax^2 + by^2 =1 \mod P$ and if so, is it computable in polynomial time? if $a$ is invertible modulo $P$ this is equivalent to $(1-by^2)/a$ is a square in $\mathcal{O}_L/P$.

2. is there any algorithm to find such $x$ and $y$ ? all such possible $(x,y)$?

3. an interesting case is: $a=b$ and $a$ is invertible modulo $P$.

Update: for $P$ above $2$, (wlog) $ord_P(a) = 0$ and $ord_P(b)=1$ there is an algorithm that outputs a solution (Algorithm 6.2)

Let $L$ be a cyclotomic field, and $P$ a prime ideal of $\mathcal{O}_L$.

1. is there any symbol for the equation $ax^2 + by^2 =1 \mod P$ and if so, is it computable in polynomial time?

2. is there any algorithm to find such $x$ and $y$ ? all such possible $(x,y)$?

3. an interesting case is: $a=b$ and $a$ is invertible modulo $P$.

Let $L$ be a cyclotomic field, and $P$ a prime ideal of $\mathcal{O}_L$.

1. is there any symbol for the equation $ax^2 + by^2 =1 \mod P$ and if so, is it computable in polynomial time? if $a$ is invertible modulo $P$ this is equivalent to $(1-by^2)/a$ is a square in $\mathcal{O}_L/P$.

2. is there any algorithm to find such $x$ and $y$ ? all such possible $(x,y)$?

3. an interesting case is: $a=b$ and $a$ is invertible modulo $P$.