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| {{infobox number
| | Wilber Berryhill is what his wife enjoys to contact him and he totally loves this title. Invoicing is my occupation. One of the issues she enjoys most is canoeing and she's been performing it for quite a while. Ohio is where my house is but my spouse desires us to transfer.<br><br>Check out my weblog [http://isaworld.pe.kr/?document_srl=392088 psychic phone] |
| | number = 65537
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| | prime = Yes
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| }}
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| [[File:Regular 65537-gon First Carlyle Circle.gif|thumb|Construction of a regular 65537-gon. See [[constructible polygon]].]]
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| '''65537''' is the integer after [[65536 (number)|65536]] and before 65538.
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| ==In mathematics==
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| 65537 is the largest known prime number of the form <math>2^{2^{n}} +1</math>, where <math>n = 4</math>. Therefore a [[regular polygon]] with 65537 sides is [[constructible polygon|constructible]] with compass and unmarked straightedge. In number
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| theory, primes of this form are known as [[Fermat number|Fermat primes]], named after the mathematician
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| [[Pierre de Fermat]]. The only known prime Fermat numbers are
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| <math>2^{2^{0}} + 1 = 2^{1} + 1 = 3,</math>
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| <math>2^{2^{1}} + 1= 2^{2} +1 = 5,</math>
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| <math>2^{2^{2}} + 1 = 2^{4} +1 = 17,</math>
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| <math>2^{2^{3}} + 1= 2^{8} + 1= 257,</math>
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| <math>2^{2^{4}} + 1 = 2^{16} + 1 = 65537.</math><ref>{{cite book |last=Conway |first=J. H. |first2=R. K. |last2=Guy |year=1996 |title=The Book of Numbers |location=New York |publisher=Springer-Verlag |page=139 |isbn=0-387-97993-X }}</ref>
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| In 1732, [[Leonhard Euler|Euler]] found that the next Fermat number is composite:
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| <math>2^{2^{5}} + 1 = 2^{32} + 1 = 4294967297 = 641 \times 6700417</math>
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| In 1880, F. Landry showed that
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| <math>2^{2^{6}} + 1 = 2^{64} + 1 = 274177 \times 67280421310721</math> | |
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| 65537 is also the 17th [[Jacobsthal-Lucas number]].
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| ==Applications==
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| 65537 is commonly used as a public exponent in the [[RSA (algorithm)|RSA]] cryptosystem. Because it is the Fermat number {{nowrap|1=F{{sub|''n''}} = 2{{sup|2{{sup|''n''}}}}}} with {{nowrap|1=''n'' = 4}}, the common shorthand is "F{{sub|4}}" or "F4".<ref>
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| {{cite web
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| | url = http://www.openssl.org/docs/apps/genrsa.html
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| | title = genrsa(1)
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| | publisher = OpenSSL Project
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| | quote = <nowiki>-F4|-3 [..] the public exponent to use, either 65537 or 3. The default is 65537.</nowiki>
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| }}
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| </ref> This value is seen as a wise compromise, since it is famously known to be prime, large enough to avoid the attacks to which small exponents make RSA vulnerable, and due to its low [[Hamming weight]] (number of 1 bits) can be computed extremely quickly on binary computers, which often support shift and increment instructions. Exponents in any base can be represented as shifts to the left in a base positional notation system, and so in binary the result is doubling - 65537 is the result of incrementing shifting 1 left by 16 places, and 16 is itself obtainable without loading a value into the register (which can be expensive when register contents approaches 64 bit), but zero and one can be derived more 'cheaply'.
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| 65537 is also used as the modulus in some [[Lehmer random number generator]]s, such as the one used by [[ZX Spectrum]], which ensures that any seed value will be coprime to it (vital to ensure the maximum period) while also allowing efficient reduction by the modulus using a bit shift and subtract.
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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:65537 (Number)}}
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| [[Category:Integers|69e04 65537]]
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Wilber Berryhill is what his wife enjoys to contact him and he totally loves this title. Invoicing is my occupation. One of the issues she enjoys most is canoeing and she's been performing it for quite a while. Ohio is where my house is but my spouse desires us to transfer.
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