Understanding Prime Numbers

A prime number, $p$, is a nat­ur­al number greater than $1$ that has ex­act­ly two dis­tinct pos­i­tive di­vi­sors: $1$ and itself. In other words, $p$ is a prime number if it cannot be writ­ten as a prod­uct of two small­er pos­i­tive in­te­gers. For­mal­ly, $p$ is prime if:

Here, the symbol $\forall$ (pro­nounced "for all") means that the state­ment ap­plies to all pos­si­ble values of $a$ and $b$ in the set of nat­ur­al num­bers $\mathbb{N}$ (pro­nounced "en"), and $\cdot$ de­notes mul­ti­pli­ca­tion.

For ex­am­ple:

• $2$ is a prime number be­cause its only di­vi­sors are $1$ and $2$.
• $3$ is prime (di­vi­sors: $1$, $3$).
• $4$ is not prime, since $2 \cdot 2 = 4$.

Prime num­bers are the build­ing blocks of the in­te­gers, since every in­te­ger greater than $1$ can be unique­ly writ­ten as a prod­uct of primes (this is known as the Fun­da­men­tal The­o­rem of Arith­metic).