Primes and the Fundamental Theorem of Arithmetic

The Atoms of the Integers

A prime is an integer greater than 1 whose only positive divisors are 1 and itself, like $2, 3, 5, 7, 11$. An integer greater than 1 that is not prime is composite. The number $1$ is neither prime nor composite because if 1 were prime, prime factorizations would lose their uniqueness.

The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be written as a product of primes, and this factorization is unique up to factor order. For example, $60 = 2^2 \cdot 3 \cdot 5$. Primes act as the unique prime fingerprints and building blocks of all integers.

Testing and Theory

Two core principles govern prime numbers:

For the primality test, if $n = ab$ with $a \le b$, then $a \le \sqrt{n}$. A factor below the square root must exist if the number is composite.

Common pitfall: Thinking 1 is prime, or checking divisors all the way up to $n$. You only need to check up to $\sqrt{n}$.