97: Difference between revisions

Revision as of 18:15, 3 July 2012

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

The number 97 is a prime number.

Property or family	Parameter values	First few numbers	Proof of satisfaction/membership/containment
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 ${\sqrt {97}}$ is between 9 and 10, verifying primality only requires us to check that the number is not divisible by any prime up to 9, i.e., any of the primes 2, 3, 5 and 7.
regular 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
Proth prime: prime of the form $k\cdot 2^{n}+1$ with $2^{n}>k$	$n=5,k=3$	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

@@ Line 14: / Line 14: @@
 | [[satisfies property::prime number]] || || {{#lst:prime number|list}} || {{divide and check up to sqrt}} Since <math>\sqrt{97}</math> is between 9 and 10, verifying primality only requires us to check that the number is not divisible by any prime up to 9, i.e., any of the primes [[2]], [[3]], [[5]] and [[7]].
 |-
 | [[satisfies property::regular prime]] || || {{#lst:regular prime|list}} ||
+|-
+| [[satisfies property::Proth prime]]: prime of the form <math>k \cdot 2^n + 1</math> with <math>2^n > k</math> || <math>n = 5, k = 3</math> || {{#lst:Proth prime|list}} ||
 |}