nLab prime number

A prime number is that which is measured [=divided by smaller number] by a monad [=unit] alone.

[Euclid, Def. 11 of Elements Book VII (~ 400-300 BC), see here]

Definition

A prime number is a natural number greater than 1 that cannot be written as a product of finitely many natural numbers (all) other than itself, hence a natural number greater than 1 that is divisible only by 1 and itself.

Equivalently, a prime number is a natural number $A$ such that $A$ is not equal to 1, and for all natural numbers $B$ and $C$, $A = B \cdot C$ implies that either $B$ is equal to 1 or $C$ is equal to 1.

This means that every natural number $n \in \mathbb{N}$ is, up to re-ordering of factors, uniquely expressed as a product of a tuple of prime numbers:

$$n \;=\; 2^{n_1} \cdot 3^{n_2} \cdot 5^{n_3} \cdot 7^{n_4} \cdot 11^{ n_5 } \cdots$$

This is called the prime factorization of $n$, proved in the fundamental theorem of arithmetic.

Notice that while the number $1 \in \mathbb{N}$ is, clearly, only divisible by one and by itself, hence might look like it deserves to be counted as a prime number, too, this would break the uniqueness of this prime factorization. In view of the general phenomenon in classifications in mathematics of some objects being too simple to be simple one could say that 1 is “too prime to be prime”.

However, historically, some authors did count 1 as a prime number, see e.g. Roegel 11.

Relation to ideals and arithmetic geometry

A number is prime if and only if it generates a maximal ideal in the rig $\mathbb{N}$ of natural numbers.

Prime numbers do not quite match the prime elements of $\mathbb{N}$, since $0$ generates a prime ideal but not a maximal ideal; instead they match the irreducible elements (Wikipedia).

From the Isbell-dual point of view, where a commutative ring such as the integers $\mathbb{Z}$ is regarded as the ring of functions on some variety, namely on Spec(Z), the fact that prime numbers $p$ correspond to maximal ideals means that they correspond to the closed points in this variety (see this Example), one also writes

$$(p) \in Spec(\mathbb{Z}) \,.$$

This dual perspective on number theory as being the geometry (algebraic geometry) over Spec(Z) is called arithmetic geometry.

function field analogy

	number fields (“function fields of curves over F1”)	function fields of curves over finite fields $\mathbb{F}_q$ (arithmetic curves)	Riemann surfaces/complex curves
affine and projective line
	$\mathbb{Z}$ (integers)	$\mathbb{F}_q[z]$ (polynomials, polynomial algebra on affine line $\mathbb{A}^1_{\mathbb{F}_q}$)	$\mathcal{O}_{\mathbb{C}}$ (holomorphic functions on complex plane)
	$\mathbb{Q}$ (rational numbers)	$\mathbb{F}_q(z)$ (rational fractions/rational function on affine line $\mathbb{A}^1_{\mathbb{F}_q}$)	meromorphic functions on complex plane
	$p$ (prime number/non-archimedean place)	$x \in \mathbb{F}_p$, where $z - x \in \mathbb{F}_q[z]$ is the irreducible monic polynomial of degree one	$x \in \mathbb{C}$, where $z - x \in \mathcal{O}_{\mathbb{C}}$ is the function which subtracts the complex number $x$ from the variable $z$
	$\infty$ (place at infinity)		$\infty$
	$Spec(\mathbb{Z})$ (Spec(Z))	$\mathbb{A}^1_{\mathbb{F}_q}$ (affine line)	complex plane
	$Spec(\mathbb{Z}) \cup place_{\infty}$	$\mathbb{P}_{\mathbb{F}_q}$ (projective line)	Riemann sphere
	$\partial_p \coloneqq \frac{(-)^p - (-)}{p}$ (Fermat quotient)	$\frac{\partial}{\partial z}$ (coordinate derivation)	“
	genus of the rational numbers = 0		genus of the Riemann sphere = 0
formal neighbourhoods
	$\mathbb{Z}/(p^n \mathbb{Z})$ (prime power local ring)	$\mathbb{F}_q [z]/((z-x)^n \mathbb{F}_q [z])$ ($n$-th order univariate local Artinian $\mathbb{F}_q$-algebra)	$\mathbb{C}[z]/((z-x)^n \mathbb{C}[z])$ ($n$-th order univariate Weil $\mathbb{C}$-algebra)
	$\mathbb{Z}_p$ (p-adic integers)	$\mathbb{F}_q[ [ z -x ] ]$ (power series around $x$)	$\mathbb{C}[ [z-x] ]$ (holomorphic functions on formal disk around $x$)
	$Spf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X$ (“$p$-arithmetic jet space” of $X$ at $p$)		formal disks in $X$
	$\mathbb{Q}_p$ (p-adic numbers)	$\mathbb{F}_q((z-x))$ (Laurent series around $x$)	$\mathbb{C}((z-x))$ (holomorphic functions on punctured formal disk around $x$)
	$\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p$ (ring of adeles)	$\mathbb{A}_{\mathbb{F}_q((t))}$ ( adeles of function field )	$\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x))$ (restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)
	$\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}})$ (group of ideles)	$\mathbb{I}_{\mathbb{F}_q((t))}$ ( ideles of function field )	$\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))$
theta functions
	Jacobi theta function
zeta functions
	Riemann zeta function	Goss zeta function
branched covering curves
	$K$ a number field ($\mathbb{Q} \hookrightarrow K$ a possibly ramified finite dimensional field extension)	$K$ a function field of an algebraic curve $\Sigma$ over $\mathbb{F}_p$	$K_\Sigma$ (sheaf of rational functions on complex curve $\Sigma$)
	$\mathcal{O}_K$ (ring of integers)		$\mathcal{O}_{\Sigma}$ (structure sheaf)
	$Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z})$ (spectrum with archimedean places)	$\Sigma$ (arithmetic curve)	$\Sigma \to \mathbb{C}P^1$ (complex curve being branched cover of Riemann sphere)
	$\frac{(-)^p - \Phi(-)}{p}$ (lift of Frobenius morphism/Lambda-ring structure)	$\frac{\partial}{\partial z}$	“
	genus of a number field	genus of an algebraic curve	genus of a surface
formal neighbourhoods
	$v$ prime ideal in ring of integers $\mathcal{O}_K$	$x \in \Sigma$	$x \in \Sigma$
	$K_v$ (formal completion at $v$)		$\mathbb{C}((z_x))$ (function algebra on punctured formal disk around $x$)
	$\mathcal{O}_{K_v}$ (ring of integers of formal completion)		$\mathbb{C}[ [ z_x ] ]$ (function algebra on formal disk around $x$)
	$\mathbb{A}_K$ (ring of adeles)		$\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x))$ (restricted product of function rings on all punctured formal disks around all points in $\Sigma$)
	$\mathcal{O}$		$\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ]$ (function ring on all formal disks around all points in $\Sigma$)
	$\mathbb{I}_K = GL_1(\mathbb{A}_K)$ (group of ideles)		$\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))$
Galois theory
	Galois group	“	$\pi_1(\Sigma)$ fundamental group
	Galois representation	“	flat connection (“local system”) on $\Sigma$
class field theory
	class field theory	“	geometric class field theory
	Hilbert reciprocity law	Artin reciprocity law	Weil reciprocity law
	$GL_1(K)\backslash GL_1(\mathbb{A}_K)$ (idele class group)	“
	$GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})$	“	$Bun_{GL_1}(\Sigma)$ (moduli stack of line bundles, by Weil uniformization theorem)
non-abelian class field theory and automorphy
	number field Langlands correspondence	function field Langlands correspondence	geometric Langlands correspondence
	$GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O})$ (constant sheaves on this stack form unramified automorphic representations)	“	$Bun_{GL_n(\mathbb{C})}(\Sigma)$ (moduli stack of bundles on the curve $\Sigma$, by Weil uniformization theorem)
	Tamagawa-Weil for number fields	Tamagawa-Weil for function fields
theta functions
	Hecke theta function		functional determinant line bundle of Dirac operator/chiral Laplace operator on $\Sigma$
zeta functions
	Dedekind zeta function	Weil zeta function	zeta function of a Riemann surface/of the Laplace operator on $\Sigma$
higher dimensional spaces
zeta functions	Hasse-Weil zeta function

Properties

Riemann hypothesis

see at Riemann hypothesis

Goldbach conjecture

see at Goldbach conjecture

Asymptotic distribution

see at prime number theorem

Specific classes of prime numbers

• Fermat prime
• Mersenne prime?
• twin prime?

 Finite sets with prime number cardinality

The following statements are equivalent for a finite set $A$:

• $A$ has a prime number cardinality

• $A$ is not a singleton, and for all finite sets $B$ and $C$ for which there exists a bijection $A \cong B \times C$, either $B$ is a singleton or $C$ is a singleton.

In dependent type theory, this is expressed for $A:\mathrm{Fin}$ as

$$\mathrm{hasPrimeCard}(A) \coloneqq \neg \mathrm{isContr}(A) \times \prod_{B:\mathrm{Fin}} \prod_{C:\mathrm{Fin}} [A \simeq B \times C] \to (\mathrm{isContr}(B) \vee \mathrm{isContr}(C))$$

Related concepts

• prime factor

• coprime integers

• Bertrand's postulate

• p-localization

• prime geodesic

• prime knot

• prime power

References

For historical discussion see

• Denis Roegel, A reconstruction of Lehmer’s table of primes (1914), 2011 (pdf)

See also

• Wikipedia, Prime number

• Neil Sloane, Sequence A000040 (The prime numbers.), The On-Line Encyclopedia of Integer Sequences, OEIS Foundation [web]