📚 Introduction:

During my math class, I developed an algorithm that generates prime numbers using prime numbers. While my algorithm for generating prime numbers may not be considered revolutionary, it is still a unique solution that showcases my ability to think creatively and analytically.

1. The set $p_k$ contains k prime numbers.

$$p_k = \{ p_0, p_1, p_2...p_k \}$$

2. Assume that $\phi$ is an integer bigger than $p_k$ and less than $p_{k+2}$. If $\phi$ is a prime, then $\phi = p_{k + 1}$

3. Let $q = \phi$, Then:

$$q\cdot P:= \left\{ (p_{0}\phi), (p_{1}\phi), (p_{2}\phi)...(p_{k}\phi) \right\}$$

4. Use the elements in $q\cdot P$ to create a $\left| P_{k} \right|\times \left| P_{k} \right|$ matrix according to:

$$\begin{pmatrix} \phi p_{0}modp_{0} & \phi p_{0}modp_{1} &\phi p_{0}modp_{2} & ... & ... & \phi p_{0}modp_{k} \\\ \phi p_{1}modp_{0} & \phi p_{1}modp_{1} &\phi p_{1}modp_{2} & ... & ... & \phi p_{1}modp_{k} \\\ \phi p_{2}modp_{0} & \phi p_{2}modp_{1} &\phi p_{2}modp_{2} & ... & ... & \phi p_{2}modp_{k} \\\ . &. & . & . & . & . \\\ . &. & . & . & . & . \\\ . &. & . & . & . & . \\\ \phi p_{k}modp_{0} & \phi p_{k}modp_{1} & . & . & . & \phi p_{k}modp_{k} & \end{pmatrix}$$

✏️ Example:

1. The set $p_4$ contains the first 4 prime numbers.

$$P_{4} = \left\{ 2, 3, 5, 7 \right\}$$

2. Let $q = 11$ . 11 is bigger than 7 and less than 13. Then:

$$q\cdot P:= \left\{(2\cdot 11), (3\cdot 11), (5\cdot 11), (7\cdot 11) \right\} = \left\{(22), (33), (55), (77) \right\}$$

3. Create a $\left| P_{4} \right|\times \left| P_{4} \right| = 4 \times 4$ matrix.

$$\begin{pmatrix} 22 mod 2&22mod3 &22mod5 &22mod7 \\\ 33mod2 & 33mod3 & 33mod5 & 33mod7 \\\ 55mod2& 55mod3& 55mod5& 55mod7\\\ 77mod2& 77mod3 & 77mod5 & 77mod7 \end{pmatrix}=\begin{pmatrix} 0&1 &2 &1 \\\ 1& 0 & 3& 5 \\\ 1& 1& 0& 6\\\ 1& 2& 2 & 0 \end{pmatrix}$$

✏️ Example:

1. $p_{14}$ generates the matrix

$$\begin{pmatrix} 0& 1& 4& 3& 6& 3 & 9 & 18 & 2 & 7 & 1 &20 & 12 & 8 \\\ 1& 0 & 1 & 1 & 9 & 11 & 5 & 8 & 3 & 25 & 17 & 30 & 18 & 12 \\\ 1 &1 & 0 & 4 & 4 & 1 &14 & 7 & 5& 3& 18& 13& 30 &20 \\\ 1 & 2 & 4 & 0& 10 & 4& 6 & 6 & 7& 10& 19& 33 & 1 &28 \\\ 1 & 1 & 2 &6 & 0 &10 & 7 & 4 &11 &24& 21 &36 &25 &1 \\\ 1 &2 &1 & 2 & 6 & 0& 16 & 3 &13 & 2& 22 &19& 37 &9 \\\ 1 & 1 & 4 & 1& 7 & 6 & 0 & 1 &17 &16 &24 &22 &20 &25 \\\ 1 & 2 & 3 & 4 & 2 & 9 & 9 & 0 &19& 23 &25 & 5 &32& 33 \\\ 1 &1 & 1 & 3 & 3 & 2 &10 &17 & 0 & 8& 27 & 8 &15 &6 \\\ 1 & 1 & 3 & 5 &10& 11 & 3& 14 & 6 & 0 &30& 31 &10 &30 \\\ 1 & 2 & 2 & 1 & 5 & 1 &12& 13 & 8 & 7 & 0 &14 &22 &38 \\\ 1 & 2 & 4 & 3 &1& 10 & 5 &10 &14 &28 & 3 & 0 &17 &19 \\\ 1 & 1 & 2 & 2 & 2 & 3 & 6 & 8 &18& 13& 5 & 3 & 0 &35 \\\ 1 & 2 & 1 & 5 & 8 & 6 & 15 & 7& 20& 20 & 6 &23 &12 &0 \end{pmatrix}$$

🧪 Experiment:

Let us look at the first prime numbers, 2 and 3. If $P = { 2, 3 }$ and $q = 4$ generates a singular matrix, change $q$ to 5. If $P = { 2, 3 }$ and $q = 5$ generates a nonsingular matrix, change $P = { 2, 3 }$ to $P = { 2, 3, 5}$. Now let us look at $P = { 2, 3, 5}$ and $q = 7$ (we can skip 6 because 6 is an even number). $P = { 2, 3, 5}$ and $q = 7$ generates a valid matrix so we can change $P = { 2, 3, 5}$ to $P = { 2, 3, 5, 7}$. $P = { 2, 3, 5, 7}$ and $q = 9$ generates a singular matrix so we can move to $q = 11$ and so on.