View source for 65,536

{{infobox number
| number        = 65536
| divisor       = 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536
}}
'''65536''' is the [[natural number]] following [[65535 (number)|65535]] and preceding [[65537 (number)|65537]].

65536 is a [[power of two]]: <math>2^{16}</math> (2 to the 16th power).

65536 is the smallest number with exactly 17 [[divisor]]s (but there are smaller numbers with more than 17 divisors; e.g., [[180 (number)|180]] has 18 divisors) {{OEIS|A005179}}.

[[File:Empty matrix 256.svg|thumb|260px|256×256 grid with 65536 squares]]

== In mathematics ==
65536 is <math>2^{2^{2^2}}</math>, so in [[tetration]] notation 65536 is&nbsp;<sup>4</sup>2.

When expressed using [[Knuth's up-arrow notation]], 65536 is
<math>
2 \uparrow 16
</math>,
which is equal to
<math>
2 \uparrow 2 \uparrow 2 \uparrow 2
</math>,
which is equivalent to
<math>
2 \uparrow\uparrow 4
</math>
or
<math>
2 \uparrow\uparrow\uparrow 3
</math>.

As <math>^{2}2</math> is also equal to 4, or <math>2 \uparrow \uparrow  2 = 4</math>,

<math>^{4}2</math> can thus be written as <math>^{^{2}2}2</math>, 
or  <math>2 \uparrow \uparrow (2 \uparrow \uparrow 2) </math>, or as the pentation, <math>2[5]3</math> (hyperoperation notation).

65536 is a [[superperfect number]] – a number such that [[divisor function|σ]](σ(''n''))&nbsp;=&nbsp;2''n''.<ref>{{MathWorld|urlname=SuperperfectNumber |title=Superperfect Number}}</ref>

A [[16-bit]] number can distinguish 65536 different possibilities. For example, [[signedness|unsigned]] [[binary number|binary notation]] exhausts all possible 16-bit codes in uniquely identifying the numbers 0 to 65535. In this scheme, 65536 is the least [[natural number]] that can ''not'' be represented with 16 bits. Conversely, it is the "first" or smallest positive integer that requires 17 bits.

65536 is the only power of 2 less than 2<sup>31000</sup> that does not contain the digits 1, 2, 4, or 8 in its [[decimal representation]].<ref>{{cite book|last=Wells|first=David|title=[[The Penguin Dictionary of Curious and Interesting Numbers]]|publisher=Penguin|year=1997|edition=revised|isbn=0-14-026149-4}}</ref>

The sum of the [[unitary divisor]]s of 65536 is prime (1 + 65536 = [[65537]], which is prime).<ref>[http://primes.utm.edu/curios/page.php/65536.html 65536] at [[Prime Pages]]</ref>

65536 is an [[untouchable number]].

== In computing ==
65,536 (2<sup>16</sup>) is the number of different values representable in a number of 16 [[bit|binary digits]] (or ''bits''), also known as an [[short integer|unsigned short integer]] in many [[computer programming]] systems. 
*A 65,536-bit integer can represent up to 2<sup>65,536</sup> (2.00352993...{{x10^|19,728}}) values.
*65,536 is the number of characters in the original [[Unicode]], and currently in a Unicode plane. 

This number is a limit in many common hardware and software implementations, some examples of which are:
* The [[Motorola 68000 family]], [[x86 architecture]], and other [[computing platform]]s have a [[Word (data type)|word]] size of 16 [[bit]]s, representing 65536 possible values. (32- or 64-bit operations are supported equally or better in modern [[microprocessor]]s.)
* Many modern CPUs allow a [[Page_(computer_memory)|memory page]] size of 64[[kibibyte|KiB]] (65536 bytes) to be configured in their [[paging|memory-management hardware]].<ref>{{cite web |url=http://static.usenix.org/publications/library/proceedings/usenix98/full_papers/ganapathy/ganapathy.pdf |title=General Purpose Operating System Support for Multiple Page Sizes |website=static.usenix.org |access-date=2012-11-02}}</ref>
* The default buffer size of a [[pipeline (Unix)|Unix pipeline]] is 64KiB (65536 bytes).
* 65536 is the maximum number of spreadsheet rows supported by [[Microsoft Excel|Excel]] 97, Excel 2000, Excel 2002 and Excel 2003. Text files that are larger than 65536 rows cannot be imported to these versions of Excel.<ref>[http://support.microsoft.com/kb/120596 Microsoft Help Q120596]</ref> (Excel 2007, 2010 and 2013 support [[1048576 (number)|1,048,576]] rows (2<sup>20</sup>)). 
* A [[16-bit]] microprocessor chip can directly access 65536 memory addresses, and the [[Highcolor|16-bit highcolor]] graphics standard supports a color palette of 65536 different colors.
* The maximum number of methods allowed in a single dex file Android application is 65536.<ref>{{Cite web|url=http://developer.android.com/tools/building/multidex.html|title=Enable multidex for apps with over 64K methods}}</ref>
* The limit for the amount of code in bytes for a non-native, non-abstract method in Java.
* The number of available ports to combine with a [[Network_layer|network address]] to create a [[network socket]].
*The maximum character limit for one message in [[WhatsApp]] is 65536.

== References ==
{{reflist}}

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[[Category:Integers]]