The modular arithmetic problem is related to the construction of a certain integer $\delta$ for some given $n$ such that $n \pm \delta$ are both positive primes. The problem is:
We will say $X \equiv (a_1, \ldots ,a_m) \mod (p_1,\ldots,p_m)$ if $X \equiv a_i \mod p_i$ for $i=1,\ldots,m$.
Suppose $X \equiv (a_1, \ldots ,a_m) \mod (p_1,\ldots,p_m)$ for primes $p_i$. Chinese Remainder Theorem tells us that there is a unique element $X$ that satisfies these congruences that lie between $0$ and $\prod p_i$. Let $n$ be a positive integer less than $\prod p_i$. How can we tell based on $X \equiv (a_1,\ldots,a_m)$ whether or not $X<n$?
Any ideas are welcome. Thank you.