**1. Two product of remainders theorems**

**Theorem**: The function is zero if, and only if, is composite, where is the remainder when divides , and is the 'th prime.

*Proof*: If is composite, then the remainder is necessarily for some which is a prime factor of . Conversely, if is for some , then is necessarily composite. The theorem follows.

**Corollary** The function is zero if, and only if, is composite, where is the remainder when divides .

**2. The function **

Fig.2.1: The above 3D graph mirrors on and the values of the function when is non-zero in the range .

Fig.2.2: The above 3D graph mirrors on and the values of the function when is non-zero in the range .

Fig.2.3: The above 3D graph mirrors on and the values of the function when is non-zero in the range .

**3. The function **

Fig.3.1: The above 3D graph mirrors on and the values of the function when is non-zero in the range .

Fig.3.2: The above 3D graph mirrors on and the values of the function when is non-zero in the range .

Fig.3.3: The above 3D graph mirrors on and the values of the function when is non-zero in the range .

**4. A product of remainders hypothesis**

**Hypothesis**: when is non-zero.

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