The integer with only 2 factors is called prime. The integar with more than 2 factors are composite. 1 is neither prime nor composite.
There are infinite primes. The most classical proof is given by Euclid based on the division algorithm.
Proof. Assume that only finite primes there are, notes them as \(p_i, i=0, 1, ..., n\), Construct a number \[ p' = \prod p_i + 1 \] Now based on the division with remainder, for any \(p_k\), \[ p' = p_k\prod_{i\neq k} p_i + 1 \] where \(\prod_{i\neq k}p_i\in\mathbb{Z}\), which shows that any \(p_k\) is not the divisor of \(p'\), which makes \(p'\) a new prime, leads to contradication.
Except the Euclid’s approach, there’re plenty of methods show the infinity of primes, we collect and introduce them in proofs_of_infinity_of_primes.
any \(n\in\mathbb{N}, n>1\) can be divided into the only form of products of prime factors. That is, \[ n = \prod p_i \]
Proof.
If we combining the same factors we have \[ n = \prod p_i^{\alpha_i} \] This is the standard factoring form of \(n\).
divisibility
Euler Function \(\varphi(n)\) represents the count of positive number less than \(n\) those coprime with \(n\). \[ \varphi(n) = n\prod\left(1-\cfrac{1}{p_i}\right) \]
Prov. Applied Eratosthenes Sieve Method using \(n\) as the limit, and with the help with inclusive-exclusive prinpicle.
The order of \(a\) to mod \(n\) is the least number \(r\) which makes \(a^r \equiv 1 \pmod m\).
Euler Theorem: If \(\gcd(a,\ m) = 1\), \[ a^{\varphi(m)} \equiv 1 \pmod m \] If the \(m=p\) is a prime number, then the coprime condition \(\gcd(a,p) = 1\) must hold, and \(\varphi(p) =p-1\). This gives the Fermat’s Little Theorem: \[ a^{p-1} \equiv 1 \pmod p \]
There are some primes with special forms that worth studying.
Fermat guesses that \(F_n = 2^{2^n} + 1\)1 is prime for all \(n\in\mathbb{N}\) by observing \(F_n = 3, 5, 17, 257, 65537\) for \(0\le n \le 4\), hence the form \(F_n\) is called Fermat number, and the prime in Fermat number is called Fermat prime. Ironically we’ve not found any prime \(F_n\) for \(n\ge 5\).
The form \(M_p = 2^p -1\)2 where \(p\) is a prime called Mersenne number, the prime in Mersenne number is called Mersenne prime. As of July 2020, 51 Mersenne primes are known. The largest known prime number, \(2^{82589933} − 1\), is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the Great Internet Mersenne Prime Search, a distributed computing project.
Eratosthenes Sieve Method
This Sieve Method based on a simple fact:
Wilson’s Theorem
For any positive \(n\in\mathbb{N}\), \(n\) is a prime if any only if \((n-1)! \equiv -1 \pmod n\).
Proof. Consider the polynomial with order \(p-2\) \[ f(x) = (x-1)(x-2)\cdots(x-(p-1)) - (x^{p-1}-1) \] The Fermat’s Little Theorem tells \(x=1, 2, \ldots, p-1\) are all the solutions of equation \(f(x)\equiv 0\). That is, the polynomial with order \(p-2\) generates \(p-1\) roots, which contradicts to Lagrange Theorem, which means the \(p(x)\) can only be a zero-polynomial mod \(p\), which indicates each cofficient could be divided by \(p\). Hence \[ \begin{align} f(p) = (p-1)! - (p^{p-1}-1) &\equiv 0 \pmod p \\ (p-1)! &\equiv -1 \pmod p \end{align} \]
Rabin-Miller Primality Test
Quadratic Test Theorem: If \(p\) is a prime, the Quadratic Congruence Equation \(x^2 \equiv 1 \pmod p\) can only be \(x=1\) or \(x = p -1\).
Prov. $$ x^2 p
(x+1)(x-1) p
x_1 , x_2p-1 $$
==Implement the RMPT==