65536 (number)

When expressed using Knuth's up-arrow notation, 65536 is $2\uparrow 16$ , which is equal to $2\uparrow 2\uparrow 2\uparrow 2$ , which is equivalent to $2\uparrow \uparrow 4$ or $2\uparrow \uparrow \uparrow 3$ .

65536 is a superperfect number - a number such that σ(σ(n))=2n.^[2]

A 16-bit number can distinguish 65536 different possibilities, such as the numbers 0..65535. In unsigned binary notation, 65536 is thus the smallest number that can not be represented by 16 bits.

65536 is the only power of 2 less than 2³¹⁰⁰⁰ that does not contain the digits 1, 2, 4 or 8 in its decimal representation.^[3]

65536 is the largest known number such that the sum of its unitary divisors is prime (1 + 65536 = 65537, which is prime).^[4]

65536 is an untouchable number.

In computing

65536 (2¹⁶) is the number of different values representable in a number of 16 binary digits (or bits), also known as an unsigned short integer in many computer programming systems. This can place a limitation on many common hardware and software implementations, some examples of which are:

The Motorola 68000 family, x86 architecture, and other computing platforms have a word size of 16 bits, representing 65536 possible values. (32- or 64-bit operations are supported equally or better in modern microprocessors.)
65536 is the maximum number of spreadsheet rows supported by 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.^[5] (Excel 2007, 2010 and 2013 support 1,048,576 rows (2²⁰)).
A 16-bit microprocessor chip can directly access 65536 memory addresses, and the 16-bit highcolor graphics standard supports a color palette of 65536 different colors.
Unreal Engine 2 only allows a view distance of 65536 UU, or Unreal Units.
The maximum number of methods allowed in a single dex file Android application is 65536.^[6]
The highest possible Trainer Identification Number in Generations I to VI of the Pokémon franchise was 65536, This was changed in Pokémon Sun & Moon.^[7]

In popular culture

There are 65536 different charts in Western geomancy.

References

↑ Sequence A005179 in OEIS
↑ Weisstein, Eric W. "Superperfect Number". MathWorld.
↑ Wells, David (1997). The Penguin Dictionary of Curious and Interesting Numbers (revised ed.). Penguin. ISBN 0-14-026149-4.
↑ 65536 at Prime Pages
↑ Microsoft Help Q120596
↑ http://developer.android.com/tools/building/multidex.html
↑ http://bulbapedia.bulbagarden.net/wiki/Trainer_ID_number

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