# Project Primus

Numeris primis ludere

### Theorem 1

An odd natural number $$n>1$$ is a prime iff $$\displaystyle\prod_{k=1}^{n-1} k \equiv n-1 \pmod {\displaystyle\sum_{k=1}^{n-1} k}.$$

### Theorem 2

Let $$p \equiv 5 \pmod 6$$ be prime then , ​$$2p+1$$ is prime iff $$2p+1 \mid 3^p-1$$ .​

### Theorem 3

Let $$p\equiv 1 \pmod 6$$ be prime and let $$5 \nmid 4p+1$$ , then $$4p+1$$ is prime iff $$4p+1 \mid 2^{2p}+1$$ .​

### Theorem 4

Let $$p_n$$ be the nth prime , then

$$p_n=1+\displaystyle\sum_{k=1}^{2 \cdot (\lfloor n\ln(n) \rfloor+1)}\left(1-\left\lfloor \frac{1}{n} \cdot \displaystyle\sum_{j=2}^k \left\lceil \frac{3-\displaystyle\sum_{i=1}^j \left\lfloor \frac{\left\lfloor \frac{j}{i} \right\rfloor}{\left\lceil \frac{j}{i} \right\rceil} \right\rfloor}{j} \right\rceil \right\rfloor\right)$$

### Theorem 5

Let $$P_j(x)=2^{-j}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{j}+\left(x+\sqrt{x^2-4}\right)^{j}\right)$$

Let $N=k\cdot 2^m-1$ such that $m>2$ , $3 \mid k$ , $0< k <2^m$ and

$\begin{cases}k \equiv 1 \pmod{10} \text{ with } m \equiv 2,3 \pmod{4} \\k \equiv 3 \pmod{10} \text{ with } m \equiv 0,3 \pmod{4} \\k \equiv 7 \pmod{10} \text{ with } m \equiv 1,2 \pmod{4} \\k \equiv 9 \pmod{10} \text{ with } m \equiv 0,1 \pmod{4}\end{cases}$

Let $$S_i=S_{i-1}^2-2$$ with $$S_0=P_k(3)$$ , then $$N$$ is prime iff $$S_{m-2} \equiv 0 \pmod N$$ .

### Theorem 6

Let $$P_j(x)=2^{-j}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{j}+\left(x+\sqrt{x^2-4}\right)^{j}\right)$$

Let $N=k\cdot 2^m-1$ such that $m>2$ , $3 \mid k$ , $0< k <2^m$ and

$\begin{cases}k \equiv 3 \pmod{42} \text{ with } m \equiv 0,2 \pmod{3} \\k \equiv 9 \pmod{42} \text{ with } m \equiv 0 \pmod{3} \\k \equiv 15 \pmod{42} \text{ with } m \equiv 1 \pmod{3} \\k \equiv 27 \pmod{42} \text{ with } m \equiv 1,2 \pmod{3} \\k \equiv 33 \pmod{42} \text{ with } m \equiv 0,1 \pmod{3} \\k \equiv 39 \pmod{42} \text{ with } m \equiv 2 \pmod{3} \end{cases}$

Let $$S_i=S_{i-1}^2-2$$ with $$S_0=P_k(5)$$ , then $$N$$ is prime iff $$S_{m-2} \equiv 0 \pmod N$$ .

### Theorem 7

Let $$P_j(x)=2^{-j}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{j}+\left(x+\sqrt{x^2-4}\right)^{j}\right)$$

Let $$N=k\cdot 2^m+1$$ such that $$m>2$$ , $$0< k <2^m$$ and

$$\begin{cases} k \equiv 1 \pmod{42} \text{ with } m \equiv 2,4 \pmod{6} \\ k \equiv 5 \pmod{42} \text{ with } m \equiv 3 \pmod{6} \\ k \equiv 11 \pmod{42} \text{ with } m \equiv 3,5 \pmod{6} \\ k \equiv 13 \pmod{42} \text{ with } m \equiv 4 \pmod{6} \\ k \equiv 17 \pmod{42} \text{ with } m \equiv 5 \pmod{6} \\ k \equiv 19 \pmod{42} \text{ with } m \equiv 0 \pmod{6} \\ k \equiv 23 \pmod{42} \text{ with } m \equiv 1,3 \pmod{6} \\ k \equiv 25 \pmod{42} \text{ with } m \equiv 0,2 \pmod{6} \\ k \equiv 29 \pmod{42} \text{ with } m \equiv 1,5 \pmod{6} \\ k \equiv 31 \pmod{42} \text{ with } m \equiv 2 \pmod{6} \\ k \equiv 37 \pmod{42} \text{ with } m \equiv 0,4 \pmod{6} \\ k \equiv 41 \pmod{42} \text{ with } m \equiv 1 \pmod{6} \end{cases}$$

Let $$S_i=S_{i-1}^2-2$$ with $$S_0=P_k(5)$$ , then $$N$$ is prime iff $$S_{m-2} \equiv 0 \pmod N$$ .

### Theorem 8

Let $$P_j(x)=2^{-j}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{j}+\left(x+\sqrt{x^2-4}\right)^{j}\right)$$

Let $$N=k\cdot 2^m+1$$ such that $$m>2$$ , $$0< k <2^m$$ and

$$\begin{cases} k \equiv 1,7 \pmod{30} \text{ with } m \equiv 0 \pmod{4} \\ k \equiv 11,23 \pmod{30} \text{ with } m \equiv 1 \pmod{4} \\ k \equiv 13,19 \pmod{30} \text{ with } m \equiv 2 \pmod{4} \\ k \equiv 17,29 \pmod{30} \text{ with } m \equiv 3 \pmod{4} \end{cases}$$

Let $$S_i=S_{i-1}^2-2$$ with $$S_0=P_k(8)$$ , then $$N$$ is prime iff $$S_{m-2} \equiv 0 \pmod N$$ .

### Theorem 9

Let $$P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$$

Let $$F_n(b)=b^{2^n}+1$$ such that $$n \ge 2$$ and $$b$$ is even number .

Let $$S_i=P_b(S_{i-1})$$ with $$S_0=P_{b}(6)$$, then If $$F_n(b)$$ is prime, then $$S_{2^n-1} \equiv 2\pmod{F_n(b)}$$.

### Theorem 10

Let $$P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$$

Let $$E_n(b)=\Large\frac{b^{2^n}+1}{2}$$ such that $$n>1$$, $$b$$ is odd number greater than one .

Let $$S_i=P_b(S_{i-1})$$ with $$S_0=P_{b}(6)$$, then If $$E_n(b)$$ is prime, then $$S_{2^n-1} \equiv 6\pmod{E_n(b)}$$.

### Theorem 11

Let $$P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$$

Let $$N_p(b)=\frac{b^p+1}{b+1}$$ , where $$p$$ is an odd prime and $$b$$ is an odd natural number greater than one .

CASE(1). $$b \equiv 1,9 \pmod{12}$$ , or $$b \equiv 3,7 \pmod{12}$$ and $$p \equiv 1 \pmod 4$$ , or $$b\equiv 5 \pmod{12}$$ and

$$p \equiv 1,7 \pmod{12}$$ , or $$b \equiv 11 \pmod{12}$$ and $$p \equiv 1,11 \pmod{12}$$ .

CASE(2). $$b \equiv 3,7 \pmod{12}$$ and $$p \equiv 3 \pmod 4$$ , or $$b\equiv 5 \pmod{12}$$ and $$p \equiv 5,11 \pmod{12}$$ , or

$$b \equiv 11 \pmod{12}$$ and $$p \equiv 5,7 \pmod{12}$$ .

Let $$S_i=P_b(S_{i-1})$$ with $$S_0=P_b(4)$$ . Suppose $$N_p(b)$$ is prime , then :

$$\bullet$$ $$S_{p-1} \equiv P_{b}(4) \pmod {N_p(b)}$$ if Case(1) holds ;

$$\bullet$$ $$S_{p-1} \equiv P_{b+2}(4) \pmod {N_p(b)}$$ if Case(2) holds ;

### Theorem 12

Let $$P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$$

Let $$M_p(a)=\frac{a^p-1}{a-1}$$ , where $$p$$ is an odd prime and $$a$$ is an odd natural number greater than one .

CASE(1). $$a \equiv 3,11 \pmod{12}$$ , or $$a \equiv 5,9 \pmod{12}$$ and $$p \equiv 1 \pmod 4$$ , or $$a\equiv 7 \pmod{12}$$ and

$$p \equiv 1,7 \pmod{12}$$ , or $$a \equiv 1 \pmod{12}$$ and $$p \equiv 1,11 \pmod{12}$$ .

CASE(2). $$a \equiv 5,9 \pmod{12}$$ and $$p \equiv 3 \pmod 4$$ , or $$a\equiv 7 \pmod{12}$$ and $$p \equiv 5,11 \pmod{12}$$ , or

$$a \equiv 1 \pmod{12}$$ and $$p \equiv 5,7 \pmod{12}$$ .

Let $$S_i=P_a(S_{i-1})$$ with $$S_0=P_a(4)$$ . Suppose $$M_p(a)$$ is prime , then :

$$\bullet$$ $$S_{p-1} \equiv P_{a}(4) \pmod {M_p(a)}$$ if Case(1) holds ;

$$\bullet$$ $$S_{p-1} \equiv P_{a-2}(4) \pmod {M_p(a)}$$ if Case(2) holds ;

### Theorem 13

Given integer $P$ , where $P>1$ let $S_k=P \cdot S_{k-1}-(2P-1) \cdot S_{k-2}+P \cdot S_{k-3}$ with $S_0=0$ , $S_1=1$ , $S_2=P-1$ .

Let $n$ be an odd natural number greater than $2$ such that $\operatorname{gcd}(P,n)=1$. Let $\left(\frac{D}{n}\right)$ be the Jacobi symbol where $D$ represents the discriminant of the characteristic polynomial: $x^3-Px^2+(2P-1)x-P$, and let $\delta(n)=n-\left(\frac{D}{n}\right)$ , then if $n$ is a prime then $S_{\delta(n)} \equiv 0 \pmod{n}$ .

### Conjecture 1

Let $$b$$ and $$n$$ be a natural numbers , $$b\geq 2$$ , $$n>2$$ and $$n \neq 9$$ . Then $$n$$ is prime if and only if

$$\displaystyle\sum_{k=1}^{n-1}\left(b^k-1\right)^{n-1} \equiv n \pmod{\frac{b^n-1}{b-1}}$$

### Conjecture 2

Let $$b$$ and $$n$$ be a natural numbers , $$b\geq 2$$ , $$n>1$$ and $$n \not\in \{4,8,9\}$$ . Then $$n$$ is prime if and only if

$$\displaystyle\sum_{k=1}^{n}\left(b^k+1\right)^{n-1} \equiv n \pmod{\frac{b^n-1}{b-1}}$$

### Conjecture 3

If $$q$$ is the smallest prime greater than $$\displaystyle\prod_{i=1}^n C_i+1$$ , where $$\displaystyle\prod_{i=1}^n C_i$$ is the product of the first $$n$$ composite numbers,

then $$q-\displaystyle\prod_{i=1}^n C_i$$ is prime.

### Conjecture 4

If $$q$$ is the greatest prime less than $$\displaystyle\prod_{i=1}^n C_i-1$$ , where $$\displaystyle\prod_{i=1}^n C_i$$ is the product of the first $$n$$ composite numbers,

then $$\displaystyle\prod_{i=1}^n C_i-q$$ is prime .

### Conjecture 5

Let $$n$$ be a natural number greater than two . Let $$r$$ be the smallest odd prime number such that $$r \nmid n$$ and $$n^2 \not\equiv 1 \pmod r$$ .

Let $$T_n(x)$$ be Chebyshev polynomial of the first kind , then $$n$$ is a prime number if and only if $$T_n(x) \equiv x^n \pmod {x^r-1,n}$$ .

### Conjecture 6

Let $$n$$ be a natural number greater than one and let $$F_{n}(x)$$ be Fibonacci polynomial , then $$n$$ is prime if and only if :

$$\displaystyle\sum_{k=0}^{n-1}F_{n}(k) \equiv -1 \pmod n$$ .

### Conjecture 7

Let $$P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$$

Let $$F_n(b)= b^{2^n}+1$$ where $$b$$ is an even natural number and $$n\ge2$$ . Let $$a$$ be a natural number greater

than two such that $$\left(\frac{a-2}{F_n(b)}\right)=-1$$ and $$\left(\frac{a+2}{F_n(b)}\right)=-1$$ where $$\left(\frac{}{}\right)$$ denotes Jacobi symbol. Let $$S_i=P_b(S_{i-1})$$

with $$S_0$$ equal to the modular $$P_{b/2}(P_{b/2}(a))\phantom{5} \text{mod} \phantom{5} F_n(b)$$. Then $$F_n(b)$$ is prime if and only if $$S_{2^n-2} \equiv 0 \pmod{F_n(b)}$$ .

### Conjecture 8

Let $$P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$$

Let $N= 4kp^{n} - 1$ where $k$ is a positive natural number , $4k<2^n$ , $p$ is a prime number and $n\ge3$ .

Let $a$ be a natural number greater than two such that $\left(\frac{a-2}{N}\right)=1$ and $\left(\frac{a+2}{N}\right)=-1$ where $\left(\frac{}{}\right)$ denotes Jacobi symbol.

Let $S_i=P_p(S_{i-1})$ with $S_0$ equal to the modular $P_{kp^2}(a)\phantom{5} \text{mod} \phantom{5} N$. Then $N$ is prime if and only if $S_{n-2} \equiv 0 \pmod{N}$ .

### Conjecture 9

Let $$P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$$

Let $N= 4kp^{n} + 1$ where $k$ is a positive natural number , $4k<2^n$ , $p$ is a prime number and $n\ge3$ .

Let $a$ be a natural number greater than two such that $\left(\frac{a-2}{N}\right)=-1$ and $\left(\frac{a+2}{N}\right)=-1$ where $\left(\frac{}{}\right)$ denotes Jacobi symbol.

Let $S_i=P_p(S_{i-1})$ with $S_0$ equal to the modular $P_{kp^2}(a)\phantom{5} \text{mod} \phantom{5} N$. Then $N$ is prime if and only if $S_{n-2} \equiv 0 \pmod{N}$ .

### Conjecture 10

Let $$P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$$

Let $$M_p(a)= \frac{a^p-1}{a-1}$$ where $$a$$ is a natural number greater than one and $$p$$ is an odd prime number . Let $$c$$ be

a natural number greater than two such that $$\left(\frac{c-2}{M_p(a)}\right)=-1$$ and $$\left(\frac{c+2}{M_p(a)}\right)=1$$ where $$\left(\frac{}{}\right)$$ denotes Jacobi

symbol. Let $$S_i=P_a(S_{i-1})$$ with $$S_0 =P_{a}(c)$$. Then , if $$M_p(a)$$ is prime then $$S_{p-1} \equiv P_{a-2}(c) \pmod{M_p(a)}$$ .

### Conjecture 11

Let $$P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$$

Let $$N_p(b)= \frac{b^p+1}{b+1}$$ where $$b$$ is a natural number greater than one and $$p$$ is an odd prime number . Let $$c$$ be

a natural number greater than two such that $$\left(\frac{c-2}{N_p(b)}\right)=-1$$ and $$\left(\frac{c+2}{N_p(b)}\right)=1$$ where $$\left(\frac{}{}\right)$$ denotes Jacobi

symbol. Let $$S_i=P_b(S_{i-1})$$ with $$S_0 =P_{b}(c)$$. Then , if $$N_p(b)$$ is prime then $$S_{p-1} \equiv P_{b+2}(c) \pmod{N_p(b)}$$ .

### Conjecture 12

Let $$P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$$

Let $$M= k \cdot b^{n}-c$$ where $$k,b,n,c$$ are natural numbers such that $$k>0$$ , $$b>1$$ , $$n>1$$ and $$c>0$$. Let $$a$$ be a

natural number greater than two such that $$\left(\frac{a-2}{M}\right)=-1$$ and $$\left(\frac{a+2}{M}\right)=1$$ where $$\left(\frac{}{}\right)$$ denotes Jacobi symbol.

Let $$S_i=P_b(S_{i-1})$$ with $$S_0$$ equal to the modular $$P_{kb}(a)\phantom{5} \text{mod} \phantom{5} M$$. Then, if $$M$$ is prime then

$$S_{n-1} \equiv P_{c-1}(a) \pmod{M}$$ .

### Conjecture 13

Let $$P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$$

Let $$N= k \cdot b^{n}+c$$ where $$k,b,n,c$$ are natural numbers such that $$k>0$$ , $$b>1$$ , $$n>1$$ and $$c>0$$ . Let $$a$$ be a

natural number greater than two such that $$\left(\frac{a-2}{N}\right)=1$$ and $$\left(\frac{a+2}{N}\right)=1$$ where $$\left(\frac{}{}\right)$$ denotes Jacobi symbol.

Let $$S_i=P_b(S_{i-1})$$ with $$S_0$$ equal to the modular $$P_{kb}(a)\phantom{5} \text{mod} \phantom{5} N$$. Then, if $$N$$ is prime then

$$S_{n-1} \equiv P_{c-1}(a) \pmod{N}$$ .

### Conjecture 14

Let $b_n=b_{n-1}+\operatorname{lcm}(\lfloor \sqrt{2} \cdot n \rfloor , b_{n-1})$ with $b_1=2$ and $n>1$ . Let $a_n=b_{n+1}/b_n-1$ .

1. Every term of this sequence $a_i$ is either prime or $1$ .

2. Every odd prime of the form $\left\lfloor \sqrt{2}\cdot n \right\rfloor$ is a term of this sequence.

3. At the first appearance of each prime of the form $\left\lfloor \sqrt{2}\cdot n \right\rfloor$ greater than $5$, it is the next prime of the given form after the largest prime that has already appeared.

### Conjecture 15

Let $b_n=b_{n-1}+\operatorname{lcm}(\lfloor \sqrt{3} \cdot n \rfloor , b_{n-1})$ with $b_1=3$ and $n>1$ . Let $a_n=b_{n+1}/b_n-1$ .

1. Every term of this sequence $a_i$ is either prime or $1$ .

2. Every odd prime of the form $\left\lfloor \sqrt{3}\cdot n \right\rfloor$ greater than $3$ is a term of this sequence.

3. At the first appearance of each prime of the form $\left\lfloor \sqrt{3}\cdot n \right\rfloor$ greater than $5$, it is the next prime of the given form after the largest prime that has already appeared.

### Conjecture 16

Let $b_n=b_{n-1}+\operatorname{lcm}(\lfloor \sqrt{n^3} \rfloor , b_{n-1})$ with $b_1=2$ and $n>1$ . Let $a_n=b_{n+1}/b_n-1$ .

1. Every term of this sequence $a_i$ is either prime or $1$ .

2. Every odd prime of the form $\lfloor \sqrt{n^3} \rfloor$ is member of this sequence .

3. At the first appearance of each prime of the form $\lfloor \sqrt{n^3} \rfloor$ greater than $5$, it is the next prime of the given form after the largest prime that has already appeared.

### Conjecture 17

Let $a$ , $b$ and $n$ be a natural numbers , $b>a>1$ , $n>2$ and $n \not\in \{4,9,25\}$ .

Then $n$ is prime iff $\displaystyle \prod_{k=1}^{n-1} \left(b^k-a\right) \equiv \frac{a^n-1}{a-1} \pmod{\frac{b^n-1}{b-1}}$

### Conjecture 18

Let $a$ , $b$ and $n$ be a natural numbers , $b>a>0$ , $n>2$ and $n \not\in \{4,9,25\}$ .

Then $n$ is prime iff $\displaystyle \prod_{k=1}^{n-1} \left(b^k+a\right) \equiv \frac{a^n+1}{a+1} \pmod{\frac{b^n-1}{b-1}}$

### Conjecture 19

Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$ .

Let $N=8kp^n-1$ such that $k>0$ , $3 \not\mid k$ , $p$ is a prime number, $p \neq 3$ , $n > 2$ and $8k < p^n$ .

Let $S_i=P_p(S_{i-1})$ with $S_0=P_{2kp^2}(4)$ , then: $N$ is a prime iff $S_{n-2} \equiv 0\pmod{N}$

### Conjecture 20

Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$ .

Let $N=8k3^n-1$ such that $n>2$ , $k>0$ , $8k < 3^n$ and

$\begin{cases} k \equiv 1 \pmod{5} \text{ with } n \equiv 0,1 \pmod{4} \\ k \equiv 2 \pmod{5} \text{ with } n \equiv 1,2 \pmod{4} \\ k \equiv 3 \pmod{5} \text{ with } n \equiv 0,3 \pmod{4} \\ k \equiv 4 \pmod{5} \text{ with } n \equiv 2,3 \pmod{4} \end{cases}$

Let $S_i=S_{i-1}^3-3S_{i-1}$ with $S_0=P_{18k}(3)$ , then $N$ is prime iff $S_{n-2} \equiv 0 \pmod N$ .

### Conjecture 21

Let $n$ be an odd natural number that is not a perfect square and let $c$ be the smallest odd prime number such that

$\left(\frac{c}{n}\right)=-1$, where $\left(\frac{}{}\right)$ denotes a Jacobi symbol . Let $T_m(x)$ be the mth Chebyshev polynomial of the first kind,

then $n$ is prime iff $T_n\left(1+\sqrt{c}\right) \equiv 1- \sqrt{c} \pmod{n}$ .

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