1. Two product of remainders theorems

Theorem: The function $f(n) = \prod_{_{k=2}}^{^{\pi(\sqrt{n})}}[\frac{n}{p_{_{{k}}}}]$ is zero if, and only if, $n$ is composite, where $0 \leq [\frac{n}{m}] < m$ is the remainder when $m$ divides $n$, and $p_{_{k}}$ is the $k$'th prime.

Proof: If $n$ is composite, then the remainder $[\frac{n}{p_{_{k}}}]$ is necessarily $0$ for some $p_{_{k}} < p_{_{\pi(\sqrt{n})}}$ which is a prime factor of $n$. Conversely, if $[\frac{n}{p_{_{k}}}]$ is $0$ for some $p_{_{k}} < p_{_{\pi(\sqrt{n})}}$, then $n$ is necessarily composite. The theorem follows.

Corollary The function $f(n) = \prod_{_{k=2}}^{^{n-1}}[\frac{n}{k}]$ is zero if, and only if, $n$ is composite, where $0 \leq [\frac{n}{m}] < m$ is the remainder when $m$ divides $n$.

2. The function $log_{_{e}}\prod_{_{k=2}}^{^{\pi(\sqrt{n})}}[\frac{n}{p_{_{{k}}}}]$ Fig.2.1: The above 3D graph mirrors on $y=0$ and $y=1$ the values of the function $log_{_{e}}{\prod_{_{k=2}}^{^{\pi(\sqrt{n})}}[\frac{n}{p_{_{{k}}}}]}$ when $\prod_{_{k=2}}^{^{\pi(\sqrt{n})}}[\frac{n}{p_{_{{k}}}}]$ is non-zero in the range $4 \leq n \leq 100$. Fig.2.2: The above 3D graph mirrors on $y=0$ and $y=1$ the values of the function $log_{_{e}}{\prod_{_{k=2}}^{^{\pi(\sqrt{n})}}[\frac{n}{p_{_{{k}}}}]}$ when $\prod_{_{k=2}}^{^{\pi(\sqrt{n})}}[\frac{n}{p_{_{{k}}}}]$ is non-zero in the range $4 \leq n \leq 1000$. Fig.2.3: The above 3D graph mirrors on $y=0$ and $y=1$ the values of the function $log_{_{e}}{\prod_{_{k=2}}^{^{\pi(\sqrt{n})}}[\frac{n}{p_{_{{k}}}}]}$ when $\prod_{_{k=2}}^{^{\pi(\sqrt{n})}}[\frac{n}{p_{_{{k}}}}]$ is non-zero in the range $4 \leq n \leq 10000000$.

3. The function $log_{_{e}}\prod_{_{k=2}}^{^{\pi(\sqrt{n})}}[\frac{n}{p_{_{{k}}}}]/\sqrt{n}$ Fig.3.1: The above 3D graph mirrors on $y=0$ and $y=1$ the values of the function $log_{_{e}}{\prod_{_{k=2}}^{^{\pi(\sqrt{n})}}[\frac{n}{p_{_{{k}}}}]/\sqrt{n}}$ when $\prod_{_{k=2}}^{^{\pi(\sqrt{n})}}[\frac{n}{p_{_{{k}}}}]$ is non-zero in the range $4 \leq n \leq 100$. Fig.3.2: The above 3D graph mirrors on $y=0$ and $y=1$ the values of the function $log_{_{e}}{\prod_{_{k=2}}^{^{\pi(\sqrt{n})}}[\frac{n}{p_{_{{k}}}}]/\sqrt{n}}$ when $\prod_{_{k=2}}^{^{\pi(\sqrt{n})}}[\frac{n}{p_{_{{k}}}}]$ is non-zero in the range $4 \leq n \leq 1000$. Fig.3.3: The above 3D graph mirrors on $y=0$ and $y=1$ the values of the function $log_{_{e}}{\prod_{_{k=2}}^{^{\pi(\sqrt{n})}}[\frac{n}{p_{_{{k}}}}]/\sqrt{n}}$ when $\prod_{_{k=2}}^{^{\pi(\sqrt{n})}}[\frac{n}{p_{_{{k}}}}]$ is non-zero in the range $4 \leq n \leq 10000000$.

4. A product of remainders hypothesis

Hypothesis: $\prod_{_{k=2}}^{^{\pi(\sqrt{n})}}[\frac{n}{p_{_{{k}}}}] = O(e^{^{\sqrt{n}}})$ when $[\frac{n}{p_{_{{k}}}}]$ is non-zero. 