11

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Summary

Factorization

The number 11 is a prime number.

Properties and families

Property or family	Parameter values	First few numbers	Proof of satisfaction/containment/membership
prime number	fifth 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
safe prime (prime that is of the form $2q+1$ for other prime $q$)	third 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
Sophie Germain prime (prime $p$ such that $2p+1$ is prime)	fourth 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
regular prime	fourth regular prime	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 11

Discrete logarithm

2 is a primitive root mod 11, 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 10 to the multiplicative group of nonzero congruence classes mod 11, 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 $n\equiv 2^{m}{\pmod {1}}1$:

Congruence class mod 11 (written as smallest positive integer)	Congruence class mod 11 (written as smallest magnitude integer)	Discrete logarithm to base 2, written as integer mod 10	Is it a primitive root mod 11 (if and only if the discrete log is relatively prime to 10)?	Is it s quadratic residue or nonresidue mod 11 (residue if discrete log is even, nonresidue if odd)
1	1	0	No	quadratic residue
2	2	1	Yes	quadratic nonresidue
3	3	8	No	quadratic residue
4	4	2	No	quadratic residue
5	5	4	No	quadratic residue
6	-5	9	Yes	quadratic nonresidue
7	-4	7	Yes	quadratic nonresidue
8	-3	3	Yes	quadratic nonresidue
9	-2	6	No	quadratic residue
10	-1	5	No	quadratic nonresidue