5
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Contents
Summary
Factorization
The number 5 is a prime number.
Properties and families
Property or family | Parameter values | First few numbers | Proof of satisfaction/membership/containment |
---|---|---|---|
prime number | it is the third prime number | 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, [SHOW MORE]View list on OEIS | divide and check |
Fermat number | , i.e., | 3, 5, 17, 257, 65537, 4294967297 View list on OEIS | plug and check |
Fermat prime (Fermat number that is also a prime number) | combine above | ||
Proth prime: prime of the form with | 3, 5, 13, 17, 41, 97, 113, [SHOW MORE]View list on OEIS | ||
safe prime (prime of the form , prime) | first safe prime | 5, 7, 11, 23, 47, 59, 83, 107, 167, [SHOW MORE]View list on OEIS | , 2 is prime |
Sophie Germain prime (prime such that is prime | third Sophie Germain prime | 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, [SHOW MORE]View list on OEIS | , 11 is prime |
regular prime | second regular prime (2 is neither regular nor irregular) | 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, [SHOW MORE]View list on OEIS | |
twin primes (greater member of pair) | first greater member of twin prime pair | 5, 7, 13, 19, 31, 43, 61, 73, 103, [SHOW MORE]View list on OEIS | 5 - 2 = 3 is a prime |
twin primes (lesser member of pair) | second lesser member of twin prime pair | 3, 5, 11, 17, 29, 41, 59, 71, 101, [SHOW MORE]View list on OEIS | 5 + 2 = 7 is a prime |
near-square prime of the form | (second prime of this form) | 2, 5, 17, 37, 101, 197, 257, [SHOW MORE]View list on OEIS |
Structure of integers mod 5
Discrete logarithm
Template: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 . The inverse of that mapping is the discrete logarithm, i.e., the discrete logarithm of is means :
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:
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 , the primitive roots are precisely and .
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 , the quadratic residues are 1 and and the quadratic nonresidues are and . 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.