29: Difference between revisions

Latest revision as of 18:41, 3 July 2012

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The number 29 is a prime number.

Property or family	Parameter values	First few numbers	Proof of satisfaction/membership/containment
prime number	tenth 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 ${\sqrt {29}}$ is between 5 and 6, verifying primality requires verifying that 29 is not divisible by any prime up to 5, i.e., it is not divisible by 2, 3, or 5.
Sophie Germain prime (prime $p$ such that $2p+1$ is prime)	sixth Sophie Germain prime	2, 3, 5, 11, 23, 29, 41, 53, 83, 89, [SHOW MORE] 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, 1013, 1019, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1439, 1451, 1481, 1499, 1511, 1559 View list on OEIS	$2(29)+1$ equals 59, which is prime

Below are some polynomials that give prime numbers for small input values, which give the value 29 for suitable input choice.

Polynomial	Degree	Some values for which it generates primes	Input value $n$ at which it generates 29
$n^{2}-n+17$	2	all numbers 1-16, because 17 is one of the lucky numbers of Euler.	4

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 The number 29 is a [[prime number]].
+===Properties and families===
+{| class="sortable" border="1"
+! Property or family !! Parameter values !! First few numbers !! Proof of satisfaction/membership/containment
+|-
+| [[satisfies property::prime number]] || tenth prime number || {{#lst:prime number|list}} || {{divide and check up to sqrt}} In this case, since <math>\sqrt{29}</math> is between 5 and 6, verifying primality requires verifying that 29 is not divisible by any prime up to 5, i.e., it is not divisible by [[2]], [[3]], or [[5]].
+|-
+| [[satisfies property::Sophie Germain prime]] (prime <math>p</math> such that <math>2p + 1</math> is prime) || sixth Sophie Germain prime || {{#lst:Sophie Germain prime|list}} || <math>2(29) + 1</math> equals [[59]], which is prime
+|}
+==Prime-generating polynomials==
+Below are some polynomials that give prime numbers for small input values, which give the value 29 for suitable input choice.
+{| class="sortable" border="1"
+! Polynomial !! Degree !! Some values for which it generates primes !! Input value <math>n</math> at which it generates 29
+|-
+| <math>n^2 - n + 17</math> || 2 || all numbers 1-16, because 17 is one of the [[lucky numbers of Euler]]. || 4
+|}