Prime Numbers

The definition, and two common exceptions

A prime numberis a natural number greater than 1 whose only positive divisors are 1 and itself. A number greater than 1 that isn't prime is composite.

0 is neither prime nor composite: every number times 0 is 0, so 0 would have infinitely many factors, and primality requires a finite, specific set of divisors.

1 is also neither: a prime needs exactly two distinct divisors (1 and itself), but 1's only divisor is 1 itself, just one.

Worked example: checking primality

Is $91$ prime?

Solution: It's enough to test divisibility by every prime up to $\sqrt{91} \approx 9.5$ (2, 3, 5, and 7), since any factor larger than the square root would have to pair with one smaller than it, already caught. $91$is odd (not divisible by 2), its digit sum 10 isn't divisible by 3, it doesn't end in 0 or 5 (not divisible by 5), but $91 = 7 \times 13$, so 91 is composite, not prime.

Twin primes

A pair of primes exactly 2 apart is called a twin prime pair, like $(17, 19)$ or $(29, 31)$. The very first few pairs are $(3,5), (5,7), (11,13), (17,19), (29,31), (41,43)$.