Prime Number
ArithmeticA prime number is a whole number greater than 1 that can only be divided evenly by 1 and itself.
Definition
A prime number is a counting number bigger than 1 that can only be divided evenly by 1 and itself. It has exactly two factors.
Example
$7$ is prime: only $1 \times 7 = 7$. But $6$ is NOT prime because $2 \times 3 = 6$, so $6$ has four factors.
Key Insight
Think of primes as numbers that cannot be arranged into a rectangle of more than one row. $7$ dots can only be arranged as $1$ row of $7$.
Definition
A prime number is a natural number greater than $1$ with exactly two positive divisors: $1$ and itself. The number $1$ is neither prime nor composite. By the Fundamental Theorem of Arithmetic, every integer greater than $1$ is either prime or a unique product of primes.
Example
$13$ is prime (divisors: $1, 13$). $15$ is composite (divisors: $1, 3, 5, 15$). Primes up to $30$: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$.
Key Insight
$2$ is the only even prime. The Sieve of Eratosthenes finds all primes up to $n$ by crossing out multiples of each prime starting at $2$.
Definition
A prime $p$ in $\mathbb{Z}$ is an irreducible element: if $p \mid ab$ then $p \mid a$ or $p \mid b$. The Prime Number Theorem states $\pi(n) \sim n / \ln(n)$ where $\pi(n)$ counts primes up to $n$. Dirichlet's theorem: for $\gcd(a,d)=1$, there are infinitely many primes of the form $a + kd$.
Example
Mersenne primes: $2^p - 1$ (e.g., $2^7 - 1 = 127$). RSA encryption relies on the hardness of factoring $n = pq$ for large primes $p, q$.
Key Insight
Primes are "atoms" of multiplication in $\mathbb{Z}$. The Riemann Hypothesis, if proved, would give the tightest known bound on the error in the Prime Number Theorem, with implications throughout analytic number theory.