7: Difference between revisions

Revision as of 22:29, 2 January 2012

This article is about a particular natural number.|View all articles on particular natural numbers

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

Discrete logarithm

Template:Discrete log facts to check against

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

@@ Line 21: / Line 21: @@
 |-
 | [[satisfies property::regular prime]] || third regular prime || {{#lst:regular prime|list}} ||
+|}
+==Discrete logarithm==
+{{discrete log facts to check against}}
+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 <math>m \mapsto 3^m</math>. The inverse of that mapping is the discrete logarithm, i.e., the discrete logarithm of <math>n</math> is <math>m</math> means <math>n \equiv 3^m \pmod 7</math>:
+{| class="sortable" border="1"
+! 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]]
 |}