65,536

Last updated
65535 65536 65537
Cardinal sixty-five thousand five hundred thirty-six
Ordinal 65536th
(sixty-five thousand five hundred thirty-sixth)
Factorization 216
Divisors 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536
Greek numeral ͵εφλϚ´
Roman numeral LXVDXXXVI, lxvdxxxvi
Binary 100000000000000002
Ternary 100222200213
Senary 12232246
Octal 2000008
Duodecimal 31B1412
Hexadecimal 1000016

65536 is the natural number following 65535 and preceding 65537.

Contents

65536 is a power of two: (2 to the 16th power).

65536 is the smallest number with exactly 17 divisors (but there are smaller numbers with more than 17 divisors; e.g., 180 has 18 divisors) (sequence A005179 in the OEIS ).

256x256 grid with 65536 squares Empty matrix 256.svg
256×256 grid with 65536 squares

In mathematics

65536 is , so in tetration notation 65536 is 42.

When expressed using Knuth's up-arrow notation, 65536 is , which is equal to , which is equivalent to or .

As is also equal to 4, or ,

can thus be written as , or , or as the pentation (hyperoperation notation).

65536 is a superperfect number – a number such that σ(σ(n)) = 2n. [1]

A 16-bit number can distinguish 65536 different possibilities. For example, unsigned 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 231000 that does not contain the digits 1, 2, 4, or 8 in its decimal representation. [2]

The sum of the unitary divisors of 65536 is prime (1 + 65536 = 65537, which is prime). [3]

65536 is an untouchable number.

In computing

65,536 (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 number is a limit in many common hardware and software implementations, some examples of which are:

References

  1. Weisstein, Eric W. "Superperfect Number". MathWorld .
  2. Wells, David (1997). The Penguin Dictionary of Curious and Interesting Numbers (revised ed.). Penguin. ISBN   0-14-026149-4.
  3. 65536 at Prime Pages
  4. "General Purpose Operating System Support for Multiple Page Sizes" (PDF). static.usenix.org. Retrieved 2012-11-02.
  5. Microsoft Help Q120596
  6. "Enable multidex for apps with over 64K methods".