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}}}}]

Prime_Function_1_100

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 .

Prime_Function_1_1000

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 .

Prime_Function_1_10000000

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}

Prime_Function_2_100

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 .

Prime_Function_2_1000

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 .

Prime_Function_2_10000000

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.

Bhupinder Singh Anand