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{{infobox number
| number        = 65537
| prime         = 6,543rd
}}
[[File:Regular 65537-gon First Carlyle Circle.gif|thumb|Construction of a [[65537-gon|regular 65537-gon]]. See [[constructible polygon]].]]
'''65537''' is the integer after [[65536 (number)|65536]] and before 65538.

==In mathematics==

65537 is the largest known prime number of the form <math>2^{2^{n}} +1</math> (<math>n = 4</math>), and is most likely the last one.<ref>{{Cite journal |last1=Boklan |first1=Kent D. |last2=Conway |first2=John H. |date=2017 |title=Expect at most one billionth of a new Fermat Prime! |journal=The Mathematical Intelligencer |volume=39 |issue=1 |pages=3–5 |arxiv=1605.01371 |doi=10.1007/s00283-016-9644-3|s2cid=119165671 }}</ref> Therefore, a [[regular polygon|regular]] [[65537-gon|polygon with 65537 sides]] is [[constructible polygon|constructible]] with compass and unmarked straightedge. [[Johann Gustav Hermes]] gave the first explicit construction of this polygon. In number
theory, primes of this form are known as [[Fermat number|Fermat primes]], named after the mathematician
[[Pierre de Fermat]]. The only known prime Fermat numbers are

<math>2^{2^{0}} + 1 = 2^{1} + 1 = 3,</math>

<math>2^{2^{1}} + 1= 2^{2} +1 = 5,</math>

<math>2^{2^{2}} + 1 = 2^{4} +1 = 17,</math>

<math>2^{2^{3}} + 1= 2^{8} + 1= 257,</math>

<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=[https://archive.org/details/bookofnumbers0000conw/page/139 139]|isbn=0-387-97993-X |url=https://archive.org/details/bookofnumbers0000conw |url-access=registration }}</ref>

In 1732, [[Leonhard Euler]] found that the next Fermat number is composite:

<math>2^{2^{5}} + 1 = 2^{32} + 1 = 4294967297 = 641 \times 6700417</math>

In 1880, {{Interlanguage link multi|Fortuné Landry|fr}} showed that

<math>2^{2^{6}} + 1 = 2^{64} + 1 = 274177 \times 67280421310721</math>

65537 is also the 17th [[Jacobsthal–Lucas number]], and currently the largest known integer ''n'' for which the number <math>10^{n} + 27</math> is a [[probable prime]].<ref>{{Cite web |url=http://mada.la.coocan.jp/nrr/prime/primedifficulty.txt |title=Sequences by difficulty of search |access-date=2014-06-14 |archive-url=https://web.archive.org/web/20140714191755/http://mada.la.coocan.jp/nrr/prime/primedifficulty.txt |archive-date=2014-07-14 |url-status=dead }}</ref>

==Applications==
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''}}}} + 1}} with {{nowrap|1=''n'' = 4}}, the common shorthand is "F{{sub|4}}" or "F4".<ref>{{cite web
    | url = https://www.openssl.org/docs/man1.0.2/apps/genrsa.html
    | title = genrsa(1)
    | publisher = OpenSSL Project
    | quote = <nowiki>-F4|-3 [..] the public exponent to use, either 65537 or 3. The default is 65537.</nowiki>
    | access-date = 2017-05-24
    | archive-date = 2017-03-13
    | archive-url = https://web.archive.org/web/20170313225729/https://www.openssl.org/docs/man1.0.2/apps/genrsa.html
    | url-status = dead
    }}</ref> This value was used in RSA mainly for historical reasons; early raw RSA implementations (without proper padding) were vulnerable to very small exponents, while use of high exponents was computationally expensive with no advantage to security (assuming proper padding).<ref>{{Cite web |url=https://crypto.stackexchange.com/questions/109/rsa-with-small-exponents |title=RSA with small exponents?}}</ref>

65537 is also used as the modulus in some [[Lehmer random number generator]]s, such as the one used by [[ZX Spectrum]],<ref>{{cite book
 |last=Vickers |first=Steve |author-link=Steve Vickers (computer scientist)
 |title=Sinclair ZX Spectrum Basic Programming
 |url=https://worldofspectrum.org/ZXBasicManual/
 |chapter=Chapter 11. Random numbers
 |pages=73–75
 |chapter-url=https://worldofspectrum.org/ZXBasicManual/zxmanchap11.html
 |publisher=Sinclair Research Ltd.
 |year=1983 |edition=2nd
 |quote=The ZX Spectrum uses p=65537 and a=75, and stores some bi-1 in memory.
 |access-date = 2022-05-26
}}</ref> 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.

==References==
{{reflist}}

{{DEFAULTSORT:65537 (Number)}}
[[Category:Integers]]