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A family of residue classes $a_i (\bmod n_i)$ with $2\leq n_1\lt\cdots\lt n_r$ is called a covering system of congruences if every integer belongs to at least one of the residue classes, that is, every integer satisfies at least one of the congruences $a_i (\bmod n_i)$. The known examples are:

$0 (\bmod 2),\quad 0 (\bmod 3),\quad 1 (\bmod 4),\quad 5 (\bmod 6),\quad 7 (\bmod 12).$

$0 (\bmod 2),\quad 0 (\bmod 3),\quad 1 (\bmod 4),\quad 3 (\bmod 8),\quad 7 (\bmod 12),\quad 23 (\bmod 24).$

My question is whether it is possible to construct a (finite with at least two different moduli, or infinite) system of congruences that covers all odd non-square integers using moduli and residue classes satisfying

$$ \left(\frac{a_i}{n_i}\right)=-1,\qquad \text{for}\ \ 1\leq i\leq r $$

where the parenthesis is the Legendre (or Jacobi ) symbol.

P.S. What if we make the condition in the question weaker as follows: whether it is possible to construct a (finite with at least two different moduli, or infinite) system of congruences that covers all odd non-square integers using moduli and residue classes satisfying

$$ \left(\frac{a_i}{n_i}\right)=-1,\ \ \text{or}\ \ 0,\qquad \text{for}\ \ 1\leq i\leq r $$

A family of residue classes $a_i (\bmod n_i)$ with $2\leq n_1\leq\cdots\leq n_r$, ($r\geq2$) is called a covering system of congruences if every integer belongs to at least one of the residue classes, that is, every integer satisfies at least one of the congruences $a_i (\bmod n_i)$. The known examples are:

$0 (\bmod 2),\quad 0 (\bmod 3),\quad 1 (\bmod 4),\quad 5 (\bmod 6),\quad 7 (\bmod 12).$

$0 (\bmod 2),\quad 0 (\bmod 3),\quad 1 (\bmod 4),\quad 3 (\bmod 8),\quad 7 (\bmod 12),\quad 23 (\bmod 24).$

My question is whether it is possible to construct a (finite with at least two different moduli, or infinite) system of congruences that covers all odd non-square integers using moduli and residue classes satisfying

$$ \left(\frac{a_i}{n_i}\right)=-1,\qquad \text{for}\ \ 1\leq i\leq r $$

where the parenthesis is the Legendre (or Jacobi ) symbol.

P.S. What if we make the condition in the question weaker as follows: whether it is possible to construct a (finite with at least two different moduli, or infinite) system of congruences that covers all odd non-square integers using moduli and residue classes satisfying

$$ \left(\frac{a_i}{n_i}\right)=-1,\ \ \text{or}\ \ 0,\qquad \text{for}\ \ 1\leq i\leq r $$

A family of residue classes $a_i (\bmod n_i)$ with $2\leq n_1\lt\cdots\lt n_r$ is called a covering system of congruences if every integer belongs to at least one of the residue classes, that is, every integer satisfies at least one of the congruences $a_i (\bmod n_i)$. The known examples are:

$0 (\bmod 2),\quad 0 (\bmod 3),\quad 1 (\bmod 4),\quad 5 (\bmod 6),\quad 7 (\bmod 12).$

$0 (\bmod 2),\quad 0 (\bmod 3),\quad 1 (\bmod 4),\quad 3 (\bmod 8),\quad 7 (\bmod 12),\quad 23 (\bmod 24).$

My question is whether it is possible to construct a (finite with at least two different moduli, or infinite) system of congruences that covers all odd non-square integers using moduli and residue classes satisfying

$$ \left(\frac{a_i}{n_i}\right)=-1,\qquad \text{for}\ \ 1\leq i\leq r $$

where the parenthesis is the Legendre (or Jacobi ) symbol.

A family of residue classes $a_i (\bmod n_i)$ with $2\leq n_1\lt\cdots\lt n_r$ is called a covering system of congruences if every integer belongs to at least one of the residue classes, that is, every integer satisfies at least one of the congruences $a_i (\bmod n_i)$. The known examples are:

$0 (\bmod 2),\quad 0 (\bmod 3),\quad 1 (\bmod 4),\quad 5 (\bmod 6),\quad 7 (\bmod 12).$

$0 (\bmod 2),\quad 0 (\bmod 3),\quad 1 (\bmod 4),\quad 3 (\bmod 8),\quad 7 (\bmod 12),\quad 23 (\bmod 24).$

My question is whether it is possible to construct a (finite with at least two different moduli, or infinite) system of congruences that covers all odd non-square integers using moduli and residue classes satisfying

$$ \left(\frac{a_i}{n_i}\right)=-1,\qquad \text{for}\ \ 1\leq i\leq r $$

where the parenthesis is the Legendre (or Jacobi ) symbol.

P.S. What if we make the condition in the question weaker as follows: whether it is possible to construct a (finite with at least two different moduli, or infinite) system of congruences that covers all odd non-square integers using moduli and residue classes satisfying

$$ \left(\frac{a_i}{n_i}\right)=-1,\ \ \text{or}\ \ 0,\qquad \text{for}\ \ 1\leq i\leq r $$

A family of residue classes $a_i (\bmod n_i)$ with $2\leq n_1\lt\cdots\lt n_r$ is called a covering system of congruences if every integer belongs to at least one of the residue classes, that is, every integer satisfies at least one of the congruences $a_i (\bmod n_i)$. The known examples are:

$0 (\bmod 2),\quad 0 (\bmod 3),\quad 1 (\bmod 4),\quad 5 (\bmod 6),\quad 7 (\bmod 12).$

$0 (\bmod 2),\quad 0 (\bmod 3),\quad 1 (\bmod 4),\quad 3 (\bmod 8),\quad 7 (\bmod 12),\quad 23 (\bmod 24).$

My question is whether it is possible to construct a (finite or infinite) system of congruences that covers all odd non-square integers using moduli and residue classes satisfying

$$ \left(\frac{a_i}{n_i}\right)=-1,\qquad \text{for}\ \ 1\leq i\leq r $$

where the parenthesis is the Legendre (or Jacobi ) symbol.

A family of residue classes $a_i (\bmod n_i)$ with $2\leq n_1\lt\cdots\lt n_r$ is called a covering system of congruences if every integer belongs to at least one of the residue classes, that is, every integer satisfies at least one of the congruences $a_i (\bmod n_i)$. The known examples are:

$0 (\bmod 2),\quad 0 (\bmod 3),\quad 1 (\bmod 4),\quad 5 (\bmod 6),\quad 7 (\bmod 12).$

$0 (\bmod 2),\quad 0 (\bmod 3),\quad 1 (\bmod 4),\quad 3 (\bmod 8),\quad 7 (\bmod 12),\quad 23 (\bmod 24).$

My question is whether it is possible to construct a (finite with at least two different moduli, or infinite) system of congruences that covers all odd non-square integers using moduli and residue classes satisfying

$$ \left(\frac{a_i}{n_i}\right)=-1,\qquad \text{for}\ \ 1\leq i\leq r $$

where the parenthesis is the Legendre (or Jacobi ) symbol.