Power of two

In mathematics, a power of two is any of the nonnegative integer powers of the number two; in other words, two times itself a certain number of times. Note that one is a power (the zeroth power) of two. Written in binary, a power of two always has the form 10000...0, just like a power of ten in the decimal system.

Because two is the base of the binary system, powers of two are important to computer science. Specifically, two to the power of n is the number of ways the bits in a binary integer of length n can be arranged, and thus numbers that are one less than a power of two denote the upper bounds of integers in binary computers (one less because 0, not 1, is used as the lower bound). As a consequence, numbers of this form show up frequently in computer software. As one example, in the video game The Legend of Zelda for the 8-bit Nintendo, one can only hold 255 rupees at one time - the result of a byte, which is 8 bits long, being used to store the number, giving a maximum value of 2⁸-1 = 255.

Powers of two also measure computer memory. A byte is eight (2³) bits, and a kilobyte (some prefer the word kibibyte) is 1,024 (2¹⁰) bytes. Nearly all processor registers have sizes that are powers of two (32 being currently used in most personal computers).

Powers of two occur in a range of other places as well. For many disk drives, at least one of the sector size, number of sectors per track, and number of tracks per surface is a power of two. The logical block size is almost always a power of two.

Numbers which are not powers of two occur in a number of situations such as video resolutions, but they are often the sum or product of only two or three powers of two, or powers of two minus one. For example, 640 = 512 + 128, and 480 = 32 × 15. Put another way, they have fairly regular bit patterns.

A prime number that is one less than a power of two is called a Mersenne prime. For example, the prime number 31 is a Mersenne prime because it is 1 less than 32 (2⁵).

The first few powers of two

20 = 1	28 = 256	216 = 65,536
21 = 2	29 = 512	217 = 131,072
22 = 4	210 = 1,024	218 = 262,144
23 = 8	211 = 2,048	219 = 524,288
24 = 16	212 = 4,096	220 = 1,048,576
25 = 32	213 = 8,192	...
26 = 64	214 = 16,384	230 = 1,073,741,824
27 = 128	215 = 32,768	240 = 1,099,511,627,776

Powers of two whose exponents are powers of two

Because modern memory cells and registers often hold a number of bits which is a power of two, the most frequent powers of two to appear are those whose exponent is also a power of two. A short list of some of these follows.

21 = 2	216 = 65536
22 = 4	232 = 4294967296
24 = 16	264 = 18446744073709551616
28 = 256	2128 = 340282366920938463463374607431768211456

Other recognizable powers of two

• 2²⁴ = 16,777,216 - the number of unique colors that can be displayed in truecolor, which is used by common computer monitors. This number is the result of using the three-channel RGB system, with 8 bits for each channel, or 24 bits in total.