13: Difference between revisions

Revision as of 18:09, 3 July 2012

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The number 13 is a prime number.

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

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

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 12: / Line 12: @@
 ! Property or family !! Parameter values !! First few numbers !! Proof of satisfaction/membership/containment
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-| [[satisfies property::prime number]] || it is the 6th prime number || {{#lst:prime number|list}} || divide and check
+| [[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]].
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 | [[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}} ||