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Def/Prime number
From SlugmathWiki
Definition of Prime Number: A prime number, in the most traditional sense, is a natural number $p$, which has the following properties
- $p$ does not equal $1$.
- If $n$ is a natural number, and $n$ divides $p$, then $n = 1$ or $n = p$.
The prime numbers are precisely the positive irreducible elements of the ring $\ZZ$.
Logical Connections
This definition logically relies on the following definitions and statements: Def/Natural number, Def/Divides
The following statements and definitions logically rely on the material of this page: Def/Legendre symbol, Def/P-group, Def/Prime counting function, Def/Prime factorization, Def/Sylow subgroup, State/Centers of p-groups are nontrivial, State/Estimate for the product of primes, State/Fermats little theorem, State/Groups of prime order are cyclic, State/Groups of prime squared order are abelian, State/Half of nonzero residues are quadratic residues, State/Natural numbers are prime or composite or zero or one, State/Nonmultiples of a prime are relatively prime to the prime, State/Primes dividing a product must divide a factor, State/Quadratic reciprocity, State/There are infinitely many prime numbers, State/There are no zero divisors mod p, State/Uniqueness of prime factorization, State/Zolotarevs lemma, and Struct/The group with three elements.
To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Basic number theory
