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Prime numbers are the indivisible building blocks of the positive integers. Multiplication can hide structure: the number $60$ may be written as $2^2\cdot 3\cdot 5$, while $97$ resists every nontrivial factorisation. The notion of a prime number isolates exactly the integers that cannot be decomposed further under multiplication, and it makes divisibility, greatest common divisor, modular arithmetic, and unique factorisation possible as systematic theories.

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In elementary arithmetic, primes play the role that atoms play in a multiplicative monoid: every positive integer is assembled from them, and the assembly is unique up to ordering. In algebra, the same idea reappears as [prime ideals](/page/Ideal), irreducible elements, valuation theory, and the arithmetic of rings. In analytic number theory, the central question becomes not what primes are, but how they are distributed among the natural numbers.

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## Definition

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The central question is whether a [natural number](/page/Natural%20Number) has any genuine multiplicative decomposition. A prime number is one that has survived every attempt to factor it into smaller positive integers: its only positive divisors are forced by the identity element and by the number itself.

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[definition: Prime Number] A natural number $p \in \mathbb{N}$ is a prime number if $p \ge 2$ and the only positive divisors of $p$ are $1$ and $p$. [/definition]

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To make that definition usable in calculations, we also need the underlying language of divisibility. Divisibility records whether a multiplication problem can be solved inside the integers, and it is the relation against which primality is measured.

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[definition: Divides] Let $a,b \in \mathbb{Z}$. We say that $a$ divides $b$, and write $a \mid b$, if there exists $k \in \mathbb{Z}$ such that \begin{align*} b = ka. \end{align*} [/definition]

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When divisors are taken in $\mathbb{Z}$ rather than in $\mathbb{N}$, this definition says that the integer divisors of a prime $p$ are exactly $-p$, $-1$, $1$, and $p$. To understand the opposite side of the classification, we next name the numbers that do admit a genuine decomposition into smaller positive factors.

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[definition: Composite Number] A natural number $n \in \mathbb{N}$ is composite if there exist natural numbers $a,b \in \mathbb{N}$ such that \begin{align*} n = ab, \qquad 1 < a < n, \qquad 1 < b < n. \end{align*} [/definition]

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The number $1$ sits outside both classes: it is the multiplicative identity, not a building block. The divisor definition also has a practical limitation: it describes $p$ by listing its own factors, but many arguments need to recognize when $p$ must divide one factor of a product $ab$. The obstruction is that a divisor of a product need not divide either factor individually; the following definition singles out numbers for which this obstruction never occurs.

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[definition: Euclidean Prime] Let $p \in \mathbb{N}$ with $p \ge 2$. We say that $p$ is a Euclidean prime if for all $a,b \in \mathbb{Z}$, \begin{align*} p \mid ab \implies p \mid a \text{ or } p \mid b. \end{align*} [/definition]

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The product-divisibility definition is deliberately separated from the divisor definition because, in more general algebraic settings, the two ideas can diverge. Over the integers they coincide, and the characterisations below explain why primality can be recognised through greatest common divisors, products, and quotient rings.

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## Equivalent Characterisations

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The first characterisation makes precise the promised comparison between ordinary primes and Euclidean primes. It says that, in $\mathbb{Z}$, having no nontrivial positive divisors is exactly the same as detecting prime factors inside products.

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[quotetheorem:9877]

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This equivalence is often used through greatest common divisors. If a prime $p$ does not divide an integer $a$, then $a$ has no nontrivial common factor with $p$. This gives a test that is often easier to use in congruence calculations.

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[quotetheorem:9878]

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This characterisation explains why primes behave so cleanly in modular arithmetic. The same gcd dichotomy is what lets a prime factor be detected inside a product, which leads to the next standard form of primality.

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[quotetheorem:729]

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Euclid's lemma is the first place where primality shows its full strength. It also explains the basic modular-arithmetic contrast without requiring any ring language. A residue modulo $n$ is a class of integers with the same remainder after division by $n$, and multiplying residues means multiplying representatives and then reducing modulo $n$.

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