7

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Summary

Factorization

The number 7 is a prime number.

Properties and families

Property or family	Parameter values	First few numbers	Proof of satisfaction/membership/containment
prime number	fourth 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
Mersenne number $M_{n}=2^{n}-1$	$n=3$, i.e., $M_{3}=2^{3}-1$		plug and check
Mersenne prime (both a prime number and a Mersenne number)	same as for Mersenne number		combine above
safe prime (odd prime such that half of that minus one is also prime)	second 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	plug and check $(7-1)/2=3$ is prime.
regular prime	third 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
twin prime (bigger member of pair)	third twin prime, second bigger member twin prime	Twin prime	7 - 2 = 5 is prime

Structure of integers mod 7

Discrete logarithm

3 is a primitive root mod 7, so we can take that as the base of the discrete logarithm. We thus get an explicit bijection from the additive group of integers mod 6 to the multiplicative group of nonzero congruence classes mod 7, given by $m\mapsto 3^{m}$. The inverse of that mapping is the discrete logarithm, i.e., the discrete logarithm of $n$ is $m$ means $n\equiv 3^{m}{\pmod {7}}$:

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

Alternatively, we could take discrete logs to base 5, which is the other primitive root. This is simply the negative of the other discrete log:

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

Primitive roots

The number of primitive roots equals the number of generators of the additive group of integers modulo 6 (= 7 - 1) which is the Euler totient function of 6, which is 2. If $a$ is a primitive root, the primitive roots are $a$ and $a^{5}=a^{-1}$.

Explicitly, the primitive roots are 3 and 5 (= -2).

Quadratic residues and nonresidues

Of the six congruence classes of invertible elements mod 7, three are quadratic residues and three are quadratic nonresidues. If $a$ is a primitive root, the quadratic nonresidues are $a,a^{3},a^{5}$, and the quadratic residues are $a^{2},a^{4},a^{6}$. Alternatively, we can obtain the quadratic residues by taking the congruence classes of $1^{2},2^{2},3^{2}$.

Explicitly, the quadratic residues are 1,2,4 and the quadratic nonresidues are 3,5,6.

Curiosities

Constructibility of regular 7-gon

7 is the smallest prime $p$ for which the regular $p$-gon is not constructible by straightedge and compass. In fact, it is the smallest natural number $n$ for which the regular $n$-gon is not constructible by straightedge and compass. This can be seen by noting that the cyclotomic extension of adjoining 7th roots is a degree six Galois extension and cannot be expressed using successive quadratic extensions.

Significance of 10 being a primitive root

If 10 is a primitive root modulo a prime $p$, then the prime $p$ is a full reptend prime in base 10, i.e., the decimal expansion of $1/p$ has a repeating block of the maximum possible length $p-1$. The condition holds for $p=7$ (note that $10\equiv 3{\pmod {7}}$ and 3 is a primitive root), and the corresponding decimal expansion of 1/7 is:

$0\cdot {\overline {142857}}$

The corresponding number:

$\!142857$

has the property that it is a cyclic number, i.e., its product with any of the numbers from 1 to 6 is obtained by cyclically permuting its digits.