Prime number

A prime number is a positive integer $p>1$ whose only positive divisors are $1$ and $p$. The first ten prime numbers are $2,3,5,7,11,13,17,19,23,29$. In contrast, $12=3\cdot 4$ is a composite number because it has a positive divisor strictly between one and itself. The integer one is neither a prime number nor a composite number: the lower bound $p>1$ is part of the definition. Equivalently, the divisor function satisfies $\tau (p)=2$ for a prime number, whereas $\tau (1)=1$ and $\tau (n)\ge 3$ for every composite number $n$.

Prime numbers are the building blocks of positive integers under multiplication. The fundamental theorem of arithmetic says that every integer greater than one is a product of prime numbers, with a unique collection of factors when their order is ignored. For example, $12=2\cdot 2\cdot 3$ and $18=2\cdot 3\cdot 3$. Their common factors reveal the greatest common divisor $6$, and the combined maximum powers reveal the least common multiple $36$. A complicated divisibility question becomes a comparison of prime exponents.

There is also a useful finite test. If a composite number $n$ equals $ab$ with $1<a\le b$, then $a^2\le n$. Therefore some prime factor is at most $\sqrt{n}$, and trial division only needs to check prime numbers up to that square root. For $97$, checking $2,3,5,7$ suffices; none divides $97$, so it is a prime number. For $91$, the same test finds $7$, giving $91=7\cdot 13$. To find many prime numbers together, the sieve of Eratosthenes reuses the work by marking multiples.

The division with remainder calculations for $97$ make the test explicit:Since $9^2<97<10^2$, these are all the prime numbers at most its square root, and every remainder is nonzero. Thus the calculation certifies a prime number, without testing the integers $8,9,10,\dots ,96$ individually. In contrast, $91=7\cdot 13+0$ gives a prime factor and proves that $91$ is a composite number.

The definition has a long history. Book VII of Euclid's Elements distinguishes prime numbers from composite numbers through their possible measures; Book IX, Proposition 20 establishes that no finite list exhausts the prime numbers. The modern argument under Euclid's theorem needs only divisibility, so it proves an infinite conclusion without estimating how frequently prime numbers occur. ^{[ref] [ref]}.