Let $p_i$ denote the $i$-th prime number. Is there any "good function" $k(n)$ such that when we know $x_i$ for which $p_n \equiv x_i \pmod{p_i}$ for all $i\leq k(n)$, it is possible to find a unique value for $p_n$.

Note that we only assume that we know the prime numbers $p_1,p_2\ldots, p_l$ where $l\leq k(n)$ and the multiset of values $x_1,x_2,\ldots,x_l$, but not sure the order in which the values occur.

Let $p_i$ denote the $i$-th prime number. Is there any "good function" $k(n)$ such that when we know $x_i$ for which $p_n \equiv x_i \pmod{p_i}$ for all $i\leq k(n)$, it is possible to find a unique value for $p_n$.

Note that we only assume that we know the prime numbers $p_1,p_2\ldots, p_l$ where $l\leq k(n)$ and the values $x_1,x_2,\ldots,x_l$.

Let $p_i$ denote the $i$-th prime number. Is there any "good function" $k(n)$ such that when we know $x_i$ for which $p_n \equiv x_i \pmod{p_i}$ for all $i\leq k(n)$, it is possible to find a unique value for $p_n$.

Note that we only assume that we know the prime numbers $p_1,p_2\ldots, p_l$ where $l\leq k(n)$ and the values $x_1,x_2,\ldots,x_l$.

Let $p_i$ denote the $i$-th prime number. Is there any "good function" $k(n)$ such that when we know $x_i$ for which $p_n \equiv x_i \pmod{p_i}$ for all $i\leq k(n)$, it is possible to find a unique value for $p_n$.

Note that we only assume that we know the prime numbers $p_1,p_2\ldots, p_l$ where $l\leq k(n)$ and the multiset of values $x_1,x_2,\ldots,x_l$, but not sure the order in which the values occur.

Let $p_i$ denote the $i$ 'th prime number. Is there any "good function" $k(n)$ such that

When we know $x_i$ for which $p_n \equiv x_i \pmod{p_i}$ for all $i\leq k(n)$, $p_n$ is uniquely determined.

Not that we only assume that we know the prime numbers $p_1,p_2\ldots, p_l$ where $l\leq k(n)$ and the values $x_1,x_2,\ldots,x_l$.

Let $p_i$ denote the $i$-th prime number. Is there any "good function" $k(n)$ such that when we know $x_i$ for which $p_n \equiv x_i \pmod{p_i}$ for all $i\leq k(n)$, it is possible to find a unique value for $p_n$.

Note that we only assume that we know the prime numbers $p_1,p_2\ldots, p_l$ where $l\leq k(n)$ and the values $x_1,x_2,\ldots,x_l$.