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 $n\equiv 2^{m}{\pmod {11}}$:

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

Primitive roots

The number of primitive roots equals the number of generators of the additive group of integers mod 10, which is the Euler totient function of 10, which is 4. Given any primitive root $a$, the primitive roots are those whose discrete log to base $a$ is relatively prime to 10, In other words, they are the numbers $a,a^{3},a^{7},a^{9}$. The explicit list of primitive roots is 2,6,7,8.

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 11 is a safe prime that is congruent to 3 mod 8.

Quadratic residues and nonresidues

Of the ten congruence classes of invertible elements mod 11, five give quadratic residues and five give quadratic nonresidues. In terms of discrete logarithms, the quadratic residues correspond to even values and the quadratic nonresidues correspond to odd values of the logarithm. Explicitly, if $a$ is a primitive root, the quadratic residues are $1,a^{2},a^{4},a^{6},a^{8}$ and the nonresidues are $a,a^{3},a^{5},a^{7},a^{9}$.

Alternatively, the quadratic residues can be computed by taking the squares of the first five natural numbers and reducing them mod 11.

Explicitly, the quadratic residues are 1,3,4,5,9 and the quadratic nonresidues are 2,6,7,8,10.

Note that in this case, all the quadratic nonresidues other than 10 (which is -1 mod 11) are primitive roots. This follows from the fact that quadratic nonresidue that is not minus one is primitive root for safe prime.