View source for 5

{{particular natural number}}

==Summary==

===Factorization===

The number 5 is a [[prime number]].

===Properties and families===

{| class="sortable" border="1"
! Property or family !! Parameter values !! First few numbers !! Proof of satisfaction/membership/containment
|-
| [[satisfies property::prime number]] || it is the third prime number || {{#lst:prime number|list}} || divide and check
|-
| [[satisfies property::Fermat number]] <math>F_n = 2^{2^n} + 1</math> || <math>n = 1</math>, i.e., <math>F_1 = 2^{2^1} + 1</math> || {{#lst:Fermat number|list}} || plug and check
|-
| [[satisfies property::Fermat prime]] ([[Fermat number]] that is also a [[prime number]]) || || {{#lst:Fermat prime|list}} || combine above
|-
| [[satisfies property::Proth prime]]: prime of the form <math>k \cdot 2^n + 1</math> with <math>2^n > k</math> || <math>n = 2, k = 1</math> || {{#lst:Proth prime|list}} || 
|-
| [[satisfies property::safe prime]] (prime of the form <math>2q + 1</math>, <math>q</math> prime) || first safe prime || {{#lst:safe prime|list}} || <math>5 = 2 \cdot 2 + 1</math>, [[2]] is prime
|-
| [[satisfies property::Sophie Germain prime]] (prime <math>p</math> such that <math>2p + 1</math> is prime || third Sophie Germain prime || {{#lst:Sophie Germain prime|list}} || <math>2 \cdot 5 + 1 = 11</math>, [[11]] is prime
|-
| [[satisfies property::regular prime]] || second regular prime (2 is neither regular nor irregular) || {{#lst:regular prime|list}} ||
|-
| [[satisfies property::twin primes]] (greater member of pair) || first greater member of twin prime pair || {{#lst:twin primes|list-greater}} || 5 - 2 = [[3]] is a prime
|-
| [[satisfies property::twin primes]] (lesser member of pair) || second lesser member of twin prime pair || {{#lst:twin primes|list-lesser}} || 5 + 2 = [[7]] is a prime
|-
| [[satisfies property::near-square prime]] of the form <math>n^2 + 1</math> || <math>n = 2</math> (second prime of this form) || {{#lst:near-square prime|plus-one-list}} ||
|-
|}

==Structure of integers mod 5==

===Discrete logarithm===

{{discrete log facts to check against}}

2 is a [[primitive root]] mod 5, so we can take it as the base of the discrete logarithm. We thus get an explicit bijection from the additive group of integers mod 4 to the multiplicative group of nonzero congruence classes mod 5, given by <math>m \mapsto 2^m</math>. The inverse of that mapping is the discrete logarithm, i.e., the discrete logarithm of <math>n</math> is <math>m</math> means <math>n \equiv 2^m \pmod 5</math>:

{| class="sortable" border="1"
! Congruence class mod 5 (written as smallest positive integer) !! Congruence class mod 5 (written as smallest magnitude integer) !! Discrete logarithm to base 2, written as integer mod 4 !! Is it a [[primitive root]] mod 5 (if and only if the discrete log is relatively prime to 4)? !! Is it s quadratic residue or nonresidue mod 5 (residue if discrete log is even, nonresidue if odd)
|-
| 1 || 1 || 0 || No || [[quadratic residue]]
|-
| 2 || 2 || 1 || Yes || [[quadratic nonresidue]]
|-
| 3 || -2 || 3 || Yes || [[quadratic nonresidue]]
|-
| 4 || -1 || 2 || No || [[quadratic residue]]
|}

We could choose an alternative discrete logarithm, where we use 3 as the base. This is simply the negative of the original discrete logarithm:

{| class="sortable" border="1"
! Congruence class mod 5 (written as smallest positive integer) !! Congruence class mod 5 (written as smallest magnitude integer) !! Discrete logarithm to base 3, written as integer mod 4 !! Is it a [[primitive root]] mod 5 (if and only if the discrete log is relatively prime to 4)? !! Is it s quadratic residue or nonresidue mod 5 (residue if discrete log is even, nonresidue if odd)
|-
| 1 || 1 || 0 || No || [[quadratic residue]]
|-
| 2 || 2 || 3 || Yes || [[quadratic nonresidue]]
|-
| 3 || -2 || 1 || Yes || [[quadratic nonresidue]]
|-
| 4 || -1 || 2 || No || [[quadratic residue]]
|}

===Primitive roots===

The ''number'' of primitive roots equals the number of generators of the additive group of integers modulo 4, which is the [[Euler totient function]] of 4, namely 2. Given any primitive root <math>a</math>, the primitive roots are precisely <math>a</math> and <math>a^3 = a^{-1}</math>.

The explicit list of primitive roots is 2 and 3.

We note the following:

* The fact that 2 is a primitive root can be deduced from the fact that [[safe prime has plus or minus two as a primitive root]], along with the fact that 5 is congruent to 5 mod 8.
* The fact that 3 is a primitive root follows from the fact that [[Fermat prime greater than three implies three is primitive root]].

===Quadratic residues and nonresidues===

Of the four congruence classes of invertible elements, there are two classes of quadratic residues and two classes of quadratic nonresidues. Given an primitive root <math>a</math>, the quadratic residues are 1 and <math>a^2</math> and the quadratic nonresidues are <math>a</math> and <math>a^3</math>. Note that it's on account of 5 being a [[Fermat prime]] that every quadratic nonresidue is a primitive root. See [[quadratic nonresidue equals primitive root for Fermat prime]].

Explicitly, the quadratic residues are 1 and 4. The quadratic nonresidues are 2 and 3.