Let $N=pq$ where $p$ and $q$ are primes of the form $4k+1$. Let $\mathbb{Z}_N$ be the set of integers modulo $N$ and $\mathbb{Z}_N^*$ be the units in $\mathbb{Z}_N$. Let $QR$ be the quadratic residues in $\mathbb{Z}_N^*$. If none of $p$ and $q$ is $5$, then show that $\mathbb{Z}_N=\{a-b: a, b \in QR\}$. That is, we need to show that $\mathbb{Z}_N$ can be expressed as difference of quadratic residues.

Actually I observed that for N being product of two primes, apart from 3 and 5, Z_N can always be represented as difference of its set of quadratic residues. I tried to prove it but I failed. That's why I asked it here.

Initially I started with primes of the form 4k+1 as -1 is a quadratic residue there. However, nothing seems to work out.