5

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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] 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271 View list on OEIS	divide and check
Fermat number $F_n=2^{2^n}+1$	$n=1$, i.e., $F_1=2^{2^1}+1$	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
safe prime (prime of the form $2q+1$, $q$ prime)	first safe prime	5, 7, 11, 23, 47, 59, 83, 107, 167, [SHOW MORE] 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459, 2579, 2819, 2879, 2903 View list on OEIS	$5=2\cdot 2+1$, 2 is prime
Sophie Germain prime (prime $p$ such that $2p+1$ is prime	third Sophie Germain prime	2, 3, 5, 11, 23, 29, 41, 53, 83, 89, [SHOW MORE] 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, 1013, 1019, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1439, 1451, 1481, 1499, 1511, 1559 View list on OEIS	$2\cdot 5+1=11$, 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] 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 239, 241, 251, 269, 277, 281, 313, 317, 331, 337, 349, 359, 367, 373, 383, 397, 419, 431 View list on OEIS

Structure of integers mod 5

Discrete logarithm

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 $m\mapsto 2^m$. The inverse of that mapping is the discrete logarithm, i.e., the discrete logarithm of $n$ is $m$ means $n\equiv 2^m(\mathrm{mod}5)$:

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