31: Difference between revisions

Latest revision as of 01:21, 23 June 2012

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

Property or family	Parameter values	First few numbers	Proof of satisfaction/membership/containment
prime number	11th 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 ${\sqrt {n}}$ . In this case, since ${\sqrt {31}}$ is between 5 and 6, verifying primality requires checking that 31 i not divisible by any prime up to 5, i.e., it is not divisible by 2, 3, or 5.
Mersenne number $M_{n}=2^{n}-1$	$n=5$ , i.e., $M_{5}=2^{5}-1$		plug and check
Mersenne prime (both a Mersenne number and a prime number)
regular prime	10th regular prime (note that 2 is neither a regular nor an irregular 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
Euclid number	number that is of the form primorial + 1	2, 3, 7, 31, 211, 2311, View list on OEIS	30 is a primorial: it is the product of the first three primes

Below are some polynomials that give prime numbers for small input values, which give the value 31 for suitable input choice.

Polynomial	Degree	Some values for which it generates primes	Input value $n$ at which it generates 31
$n^{2}-n+11$	2	all numbers 1-10, because 11 is one of the lucky numbers of Euler.	5
$2n^{2}+29$	2	all numbers 0-28	1

Number	Prime factorization	What's interesting about it
341	11 times 31	smallest Poulet number (also called Sarrus number), i.e., smallest Fermat pseudoprime to base 2
2821	7 times 13 times 31	one of the Carmichael numbers, i.e., absolute pseudoprimes

@@ Line 12: / Line 12: @@
 ! Property or family !! Parameter values !! First few numbers !! Proof of satisfaction/membership/containment
 |-
-| [[satisfies property::prime number]] || 11th prime number || {{#lst:prime number|list}} || divide and check
+| [[satisfies property::prime number]] || 11th prime number || {{#lst:prime number|list}} || {{divide and check up to sqrt}} In this case, since <math>\sqrt{31}</math> is between 5 and 6, verifying primality requires checking that 31 i not divisible by any prime up to 5, i.e., it is not divisible by [[2]], [[3]], or [[5]].
 |-
 | [[satisfies property::Mersenne number]] <math>M_n = 2^n - 1</math> || <math>n = 5</math>, i.e., <math>M_5 = 2^5 - 1</math> || {{#lst:Mersenne number|list}} || plug and check
@@ Line 20: / Line 20: @@
 | [[satisfies property::regular prime]] || 10th regular prime (note that 2 is neither a regular nor an irregular prime) || {{#lst:regular prime|list}} ||
 |-
-| [[satisfies property::Euclid prime]] || prime that is of the form [[primorial]] + 1 || {{#lst:Euclid prime|list}} || [[30]] is a primorial: it is the product of the first three primes
+| [[satisfies property::Euclid number]] || number that is of the form [[primorial]] + 1 || {{#lst:Euclid number|list}} || [[30]] is a primorial: it is the product of the first three primes
 |}