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prime number in nLab

Contents

Context

Arithmetic

number theory

number

natural number, integer number, rational number, real number, irrational number, complex number, quaternion, octonion, adic number, cardinal number, ordinal number, surreal number

arithmetic

arithmetic geometry, function field analogy

Arakelov geometry

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 AA such that AA is not equal to 1, and for all natural numbers BB and CC, A=B⋅CA = B \cdot C implies that either BB is equal to 1 or CC is equal to 1.

This means that every natural number n∈ℕn \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⋅3 n 2⋅5 n 3⋅7 n 4⋅11 n 5⋯ 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 nn.

Notice that while the number 1∈ℕ1 \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 ℕ\mathbb{N} of natural numbers.

Prime numbers do not quite match the prime elements of ℕ\mathbb{N}, since 00 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 ℤ\mathbb{Z} is regarded as the ring of functions on some variety, namely on Spec(Z), the fact that prime numbers pp correspond to maximal ideals means that they correspond to the closed points in this variety (see this Example), one also writes

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

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

Properties

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 AA:

AA has a prime number cardinality
AA is not a singleton, and for all finite sets BB and CC for which there exists a bijection A≅B×CA \cong B \times C, either BB is a singleton or CC is a singleton.

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

hasPrimeCard(A)≔¬isContr(A)×∏ B:Fin∏ C:Fin[A≃B×C]→(isContr(B)∨isContr(C))\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))

References

For historical discussion see

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