13: Difference between revisions

Latest revision as of 18:12, 3 July 2012

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Summary

Factorization

The number 13 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 6th 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	A natural number $n>1$ is prime if and only if $n$ is not divisible by any prime less than or equal to ${\sqrt {n}}$ . Since ${\sqrt {13}}$ is between 3 and 4, we only need to check divisibility by primes less than or equal to 3, i.e., we need to verify that 13 is not divisible by the primes 2 and 3.
Proth prime: prime of the form $k\cdot 2^{n}+1$ with $2^{n}>k$	$n=2,k=3$	3, 5, 13, 17, 41, 97, 113, [SHOW MORE] 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1409, 1601, 2113, 2689, 2753, 3137, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857, 10369, 10753, 11393, 11777, 12161, 12289, 13313 View list on OEIS
regular prime	fifth 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

Polynomials

Prime-generating polynomials

Below are some polynomials that give prime numbers for small input values, which give the value 13 for suitable input choice.

Polynomial	Degree	Some values for which it generates primes	Input value $n$ at which it generates 13
$n^{2}-n+11$	2	all numbers 1-10, because 11 is one of the lucky numbers of Euler.	2

Irreducible polynomials by Cohn's irreducibility criterion

By Cohn's irreducibility criterion, we know that if we write 13 in any base greater than or equal to 2, the corresponding polynomial is irreducible. We list here all the irreducible polynomials:

Base $b$	13 in base $b$	Corresponding irreducible polynomial
2	1101	$x^{3}+x^{2}+1$
3	111	$x^{2}+x+1$
4	31	$3x+1$
5	23	$2x+3$
6	21	$2x+1$
7	16	$x+6$
8	15	$x+5$
9	14	$x+4$
10	13	$x+3$
11	12	$x+2$
12	11	$x+1$

Multiples

Interesting multiples

Number	Prime factorization	What's interesting about it
1105	5 times 13 times 17	second Carmichael number, i.e., absolute pseudoprime
1729	7 times 13 times 19	third Carmichael number, i.e., absolute pseudoprime
2821	7 times 13 times 31	fifth Carmichael number, i.e., absolute pseudoprime

@@ Line 7: / Line 7: @@
 The number 13 is a [[satisfies property::prime number]].
-==Prime-generating polynomials==
+===Properties and families===
+{| class="sortable" border="1"
+! Property or family !! Parameter values !! First few numbers !! Proof of satisfaction/membership/containment
+|-
+| [[satisfies property::prime number]] || it is the 6th prime number || {{#lst:prime number|list}} || {{divide and check up to sqrt}} Since <math>\sqrt{13}</math> is between [[3]] and [[4]], we only need to check divisibility by primes less than or equal to 3, i.e., we need to verify that 13 is not divisible by the primes [[2]] and [[3]].
+|-
+| [[satisfies property::Proth prime]]: prime of the form <math>k \cdot 2^n + 1</math> with <math>2^n > k</math> || <math>n = 2, k = 3</math> || {{#lst:Proth prime|list}} ||
+|-
+| [[satisfies property::regular prime]] || fifth regular prime (2 is neither regular nor irregular) || {{#lst:regular prime|list}} ||
+|}
+==Polynomials==
+===Prime-generating polynomials===
 Below are some polynomials that give prime numbers for small input values, which give the value 13 for suitable input choice.
@@ Line 15: / Line 29: @@
 |-
 | <math>n^2 - n + 11</math> || 2 || all numbers 1-10, because 11 is one of the [[lucky numbers of Euler]]. || 2
+|}
+===Irreducible polynomials by Cohn's irreducibility criterion===
+By [[Cohn's irreducibility criterion]], we know that if we write 13 in any base greater than or equal to 2, the corresponding polynomial is irreducible. We list here all the irreducible polynomials:
+{| class="sortable" border="1"
+! Base <math>b</math> !! 13 in base <math>b</math> !! Corresponding irreducible polynomial
+|-
+| 2 || 1101 || <math>x^3 + x^2 + 1</math>
+|-
+| 3 || 111 || <math>x^2 + x + 1</math>
+|-
+| 4 || 31 || <math>3x + 1</math>
+|-
+| 5 || 23 || <math>2x + 3</math>
+|-
+| 6 || 21 || <math>2x + 1</math>
+|-
+| 7 || 16 || <math>x + 6</math>
+|-
+| 8 || 15 || <math>x + 5</math>
+|-
+| 9 || 14 || <math>x + 4</math>
+|-
+| 10 || 13 || <math>x + 3</math>
+|-
+| 11 || 12 || <math>x + 2</math>
+|-
+| 12 || 11 || <math>x + 1</math>
+|}
+==Multiples==
+===Interesting multiples===
+{| class="sortable" border="1"
+! Number !! Prime factorization !! What's interesting about it
+|-
+| [[1105]] || [[5]] times [[13]] times [[17]] || second [[Carmichael number]], i.e., absolute pseudoprime
+|-
+| [[1729]] || [[7]] times [[13]] times [[19]] || third [[Carmichael number]], i.e., absolute pseudoprime
+|-
+| [[2821]] || [[7]] times [[13]] times [[31]] || fifth [[Carmichael number]], i.e., absolute pseudoprime
 |}