![Regular 65537-gon First Carlyle Circle](https://upload.wikimedia.org/wikipedia/commons/2/27/Regular_65537-gon_First_Carlyle_Circle.gif)
65537 is the integer after
65536
65536 is the natural number following 65535 and preceding 65537.
65536 is a power of two: 2^ (2 to the 16th power).
65536 is the smallest number with ''exactly'' 17 divisors.
In mathematics
65536 is 2^, so in tetration notation 65536 is ...
and before 65538.
In mathematics
65537 is the largest known prime number of the form
(
). Therefore, a
regular polygon with 65537 sides is
constructible with compass and unmarked straightedge.
Johann Gustav Hermes
Johann Gustav Hermes (20 June 1846 – 8 June 1912) was a German mathematician. Hermes is known for completion of a polygon with 65,537 sides.
Early life
On 20 June 1846, Hermes was born in Königsberg, a former German city (presently Kalinin ...
gave the first explicit construction of this polygon. In number
theory, primes of this form are known as
Fermat primes
In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form
:F_ = 2^ + 1,
where ''n'' is a non-negative integer. The first few Fermat numbers are:
: 3, 5, 17, 257, 65537, 4294967 ...
, named after the mathematician
Pierre de Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
. The only known prime Fermat numbers are
In 1732,
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
found that the next Fermat number is composite:
In 1880, showed that
65537 is also the 17th
Jacobsthal–Lucas number In mathematics, the Jacobsthal numbers are an integer sequence named after the German mathematician Ernst Jacobsthal. Like the related Fibonacci numbers, they are a specific type of Lucas sequence U_n(P,Q) for which ''P'' = 1, and ''Q' ...
, and currently the largest known integer ''n'' for which the number
is a
probable prime
In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers. Different types of probable primes have different specific con ...
.
Applications
65537 is commonly used as a public exponent in the
RSA cryptosystem. Because it is the Fermat number with , the common shorthand is "F" or "F4". 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).
65537 is also used as the modulus in some
Lehmer random number generator The Lehmer random number generator (named after D. H. Lehmer), sometimes also referred to as the Park–Miller random number generator (after Stephen K. Park and Keith W. Miller), is a type of linear congruential generator (LCG) that oper ...
s, such as the one used by
ZX Spectrum
The ZX Spectrum () is an 8-bit computing, 8-bit home computer that was developed by Sinclair Research. It was released in the United Kingdom on 23 April 1982, and became Britain's best-selling microcomputer.
Referred to during development as t ...
,
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
{{DEFAULTSORT:65537 (Number)
Integers