17: Difference between revisions

Revision as of 22:13, 2 January 2012

Summary

Factorization

The number 17 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 7th prime number	2,3,5,7,11,13,17,19,23,29,31, ... (never stops, infinitude of primes)	divide and check
Fermat number, Fermat prime	$F_{2}$ , where $F_{n} = 2^{2^{n}} + 1$ , starts $n = 0$	3,5,17,257,65537	plug and check
regular prime	sixth 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 17

Discrete logarithm

Template:Fermat prime discrete log facts to check against

We can take 3 to be a primitive root mod 17, i.e., a generator for the multiplicative group of integers mod 17. With this, the discrete logarithm table from the multiplicative group mod 17 to the additive group mod 16 looks as follows:

Fill this in later

@@ Line 1: / Line 1: @@
 ==Summary==
-===Properties satisfied and families it is a member of===
+===Factorization===
+The number 17 is a [[prime number]].
+===Properties and families===
 {| class="sortable" border="1"
-! Property or family !! Parameter values (if applicable) !! First few natural numbers satisfying the property
+! Property or family !! Parameter values !! First few numbers !! Proof of satisfaction/membership/containment
+|-
+| [[satisfies property::prime number]] || it is the 7th prime number || 2,3,5,7,11,13,'''17''',19,23,29,31, ... (never stops, [[infinitude of primes]]) || divide and check
 |-
-| [[satisfies property::prime number]] || it is the 7th prime number || 2,3,5,7,11,13,'''17''',19,23,29,31, ... (never stops, [[infinitude of primes]])
+| [[satisfies property::Fermat number]], [[satisfies property::Fermat prime]] || <math>F_2</math>, where <math>F_n = 2^{2^n} + 1</math>, starts <math>n = 0</math> || 3,5,'''17''',257,65537 || plug and check
 |-
-| [[satisfies property::Fermat number]], [[satisfies property::Fermat prime]] || <math>F_2</math>, where <math>F_n = 2^{2^n} + 1</math>, starts <math>n = 0</math> || 3,5,'''17''',257,65537
+| [[satisfies property::regular prime]] || sixth regular prime (2 is neither regular nor irregular) || {{#lst:regular prime|list}} ||
 |}