Let $E: y^2 = x^3 + ax + b$ be an elliptic curve over $\mathbb{Q}$ with $a,b \in \mathbb{Z}$. Recall that any rational point $P = (x,y)$ can be written uniquely as $P = (u/d^2, v/d^3)$ with $u,v,d \in \mathbb{Z}, d > 0, \gcd(u,d) = \gcd(v,d) = 1$. We write $d(P)$ for the number appearing in the denominator.

The sequence $d(nP)$ for $n \in \mathbb{N}$ is the elliptic divisibility sequence associated to $P$. My question is about existence of quadratic non-residues in these sequences modulo all but finitely many primes $p$.

Let $P$ be a point of infinite order. Does there exists a finite set of primes $S$ such that for all $p \notin S$, there is $n \in \mathbb{N}$ such that $$\left( \frac{d(nP)}{p}\right) = -1 \quad ? \quad (*)$$

Remarks:

1. It is known that the sequence $d(nP)$ contains only finitely many squares, by Theorem 1.1 of

   Everest, Reynolds, Stevens - On the denominators of rational points on elliptic curves.

   In particular, for all but finitely many $n$, one finds that $d(nP)$ is a quadratic non-residue modulo half of all primes. I want to know that as one varies $n$ and takes all these primes together, one only excludes finitely many primes.

2. There is a slightly different definition of elliptic divisibility sequences in terms of division polynomials and recurrence sequences, which can differ from the above sequence by a sign and some small primes (this is the original definition due to Ward). I'm also interested in the analogous problem for such sequences.

Let $E: y^2 = x^3 + ax + b$ be an elliptic curve over $\mathbb{Q}$ with $a,b \in \mathbb{Z}$. Recall that any rational point $P = (x,y)$ can be written uniquely as $P = (u/d^2, v/d^3)$ with $u,v,d \in \mathbb{Z}, d > 0, \gcd(u,d) = \gcd(v,d) = 1$. We write $d(P)$ for the number appearing in the denominator.

The sequence $d(nP)$ for $n \in \mathbb{N}$ is the elliptic divisibility sequence associated to $P$. My question is about existence of quadratic non-residues in these sequences modulo all but finitely many primes $p$.

Let $P$ be a point of infinite order. Does there exists a finite set of primes $S$ such that for all $p \notin S$, there is $n \in \mathbb{N}$ such that $$\left( \frac{d(nP)}{p}\right) = -1 \quad ? \quad (*)$$

Remarks:

1. It is known that the sequence $d(nP)$ contains only finitely many squares, by Theorem 1.1 of

   Everest, Reynolds, Stevens - On the denominators of rational points on elliptic curves.

   In particular, for all but finitely many $n$, one finds that $d(nP)$ is a quadratic non-residue modulo half of all primes. I want to know that as one varies $n$ and takes all these primes together, one only excludes finitely many primes.

2. There is a slightly different definition of elliptic divisibility sequences in terms of division polynomials and recurrence sequences, which can differ from the above sequence by a sign and some small primes (this is the original definition of Ward). I'm also interested in the analogous problem for such sequences.

Let $E: y^2 = x^3 + ax + b$ be an elliptic curve over $\mathbb{Q}$ with $a,b \in \mathbb{Z}$. Recall that any rational point $P = (x,y)$ can be written uniquely as $P = (u/d^2, v/d^3)$ with $u,v,d \in \mathbb{Z}, d > 0, \gcd(u,d) = \gcd(v,d) = 1$. We write $d(P)$ for the corresponding number in the denominator.

The sequence $d(nP)$ for $n \in \mathbb{N}$ is the elliptic divisibility sequence associated to $P$. My question is about existence of quadratic non-residues in these sequences modulo all but finitely many primes $p$.

Let $P$ be a point of infinite order. Does there exists a finite set of primes $S$ such that for all $p \notin S$, there is $n \in \mathbb{N}$ such that $$\left( \frac{d(nP)}{p}\right) = -1 \quad ? \quad (*)$$

Remarks:

1. It is known that the sequence $d(nP)$ contains only finitely many squares, by Theorem 1.1 of

   Everest, Reynolds, Stevens - On the denominators of rational points on elliptic curves.

   In particular, for all but finitely many $n$, one finds that $d(nP)$ is a quadratic non-residue modulo half of all primes. I want to know that as one varies $n$ and takes all these primes together, one only excludes finitely many primes.

2. There is a slightly different definition of elliptic divisibility sequences in terms of division polynomials and recurrence sequences, which can differ from the above sequence by a sign and some small primes (this is the original definition due to Ward). I'm also interested in the analogous problem for such sequences.

Let $E: y^2 = x^3 + ax + b$ be an elliptic curve over $\mathbb{Q}$ with $a,b \in \mathbb{Z}$. Recall that any rational point $P = (x,y)$ can be written uniquely as $P = (u/d^2, v/d^3)$ with $u,v,d \in \mathbb{Z}, d > 0, \gcd(u,d) = \gcd(v,d) = 1$. We write $d(P)$ for the number appearing in the denominator.

The sequence $d(nP)$ for $n \in \mathbb{N}$ is the elliptic divisibility sequence associated to $P$. My question is about existence of quadratic non-residues in these sequences modulo all but finitely many primes $p$.

Let $P$ be a point of infinite order. Does there exists a finite set of primes $S$ such that for all $p \notin S$, there is $n \in \mathbb{N}$ such that $$\left( \frac{d(nP)}{p}\right) = -1 \quad ? \quad (*)$$

Remarks:

1. It is known that the sequence $d(nP)$ contains only finitely many squares, by Theorem 1.1 of

   Everest, Reynolds, Stevens - On the denominators of rational points on elliptic curves.

   In particular, for all but finitely many $n$, one finds that $d(nP)$ is a quadratic non-residue modulo half of all primes. I want to know that as one varies $n$ and takes all these primes together, one only excludes finitely many primes.

2. There is a slightly different definition of elliptic divisibility sequences in terms of division polynomials and recurrence sequences, which can differ from the above sequence by a sign and some small primes (this is the original definition due to Ward). I'm also interested in the analogous problem for such sequences.