97: Difference between revisions

Latest revision as of 18:16, 3 July 2012

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The number 97 is a prime number.

Property or family	Parameter values	First few numbers	Proof of satisfaction/membership/containment
prime number	It is the 25th 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

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 ! Property or family !! Parameter values !! First few numbers !! Proof of satisfaction/membership/containment
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-| [[satisfies property::prime number]] || || {{#lst:prime number|list}} || divide and check
+| [[satisfies property::prime number]] || It is the 25th 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]].
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 | [[satisfies property::regular prime]] || || {{#lst:regular prime|list}} ||
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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 = 5, k = 3</math> || {{#lst:Proth prime|list}} ||
 |}