Power of two

For other uses, see Power of two (disambiguation).

A power of two is a number of the form 2ⁿ where n is an integer, that is, the result of exponentiation with number two as the base and integer n as the exponent.

Visualization of powers of two from 1 to 1024 (2⁰ to 2¹⁰)

In a context where only integers are considered, n is restricted to non-negative values,^[1] so there are 1, 2, and 2 multiplied by itself a certain number of times.^[2]

The first ten powers of 2 for non-negative values of n are:

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... (sequence A000079 in the OEIS)

Base of the binary numeral system

Because two is the base of the binary numeral system, powers of two are common in computer science. Written in binary, a power of two always has the form 100...000 or 0.00...001, just like a power of 10 in the decimal system.

Computer science

Two to the exponent of n, written as 2ⁿ, is the number of ways the bits in a binary word of length n can be arranged. A word, interpreted as an unsigned integer, can represent values from 0 (000...000₂) to 2ⁿ − 1 (111...111₂) inclusively. Corresponding signed integer values can be positive, negative and zero; see signed number representations. Either way, one less than a power of two is often the upper bound of an integer in binary computers. As a consequence, numbers of this form show up frequently in computer software. As an example, a video game running on an 8-bit system might limit the score or the number of items the player can hold to 255—the result of using a byte, which is 8 bits long, to store the number, giving a maximum value of 2⁸ − 1 = 255. For example, in the original Legend of Zelda the main character was limited to carrying 255 rupees (the currency of the game) at any given time, and the video game Pac-Man famously has a kill screen at level 256.

Powers of two are often used to measure computer memory. A byte is now considered eight bits (an octet), resulting in the possibility of 256 values (2⁸). (The term byte once meant (and in some cases, still means) a collection of bits, typically of 5 to 32 bits, rather than only an 8-bit unit.) The prefix kilo, in conjunction with byte, may be, and has traditionally been, used, to mean 1,024 (2¹⁰). However, in general, the term kilo has been used in the International System of Units to mean 1,000 (10³). Binary prefixes have been standardized, such as kibi (Ki) meaning 1,024. Nearly all processor registers have sizes that are powers of two, 32 or 64 being very common.

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 that 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 = 32 × 20, and 480 = 32 × 15. Put another way, they have fairly regular bit patterns.

Mersenne and Fermat primes

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). Similarly, a prime number (like 257) that is one more than a positive power of two is called a Fermat prime—the exponent itself is a power of two. A fraction that has a power of two as its denominator is called a dyadic rational. The numbers that can be represented as sums of consecutive positive integers are called polite numbers; they are exactly the numbers that are not powers of two.

Euclid's Elements, Book IX

The geometric progression 1, 2, 4, 8, 16, 32, ... (or, in the binary numeral system, 1, 10, 100, 1000, 10000, 100000, ... ) is important in number theory. Book IX, Proposition 36 of Elements proves that if the sum of the first n terms of this progression is a prime number (and thus is a Mersenne prime as mentioned above), then this sum times the nth term is a perfect number. For example, the sum of the first 5 terms of the series 1 + 2 + 4 + 8 + 16 = 31, which is a prime number. The sum 31 multiplied by 16 (the 5th term in the series) equals 496, which is a perfect number.

Book IX, Proposition 35, proves that in a geometric series if the first term is subtracted from the second and last term in the sequence, then as the excess of the second is to the first—so is the excess of the last to all those before it. (This is a restatement of our formula for geometric series from above.) Applying this to the geometric progression 31, 62, 124, 248, 496 (which results from 1, 2, 4, 8, 16 by multiplying all terms by 31), we see that 62 minus 31 is to 31 as 496 minus 31 is to the sum of 31, 62, 124, 248. Therefore, the numbers 1, 2, 4, 8, 16, 31, 62, 124 and 248 add up to 496 and further these are all the numbers that divide 496. For suppose that p divides 496 and it is not amongst these numbers. Assume p q is equal to 16 × 31, or 31 is to q as p is to 16. Now p cannot divide 16 or it would be amongst the numbers 1, 2, 4, 8 or 16. Therefore, 31 cannot divide q. And since 31 does not divide q and q measures 496, the fundamental theorem of arithmetic implies that q must divide 16 and be amongst the numbers 1, 2, 4, 8 or 16. Let q be 4, then p must be 124, which is impossible since by hypothesis p is not amongst the numbers 1, 2, 4, 8, 16, 31, 62, 124 or 248.

Table of values

(sequence A000079 in the OEIS)

n	2^n		n	2^n		n	2^n		n	2^n
0	1		16	65,536		32	4,294,967,296		48	281,474,976,710,656
1	2		17	131,072		33	8,589,934,592		49	562,949,953,421,312
2	4		18	262,144		34	17,179,869,184		50	1,125,899,906,842,624
3	8		19	524,288		35	34,359,738,368		51	2,251,799,813,685,248
4	16		20	1,048,576		36	68,719,476,736		52	4,503,599,627,370,496
5	32		21	2,097,152		37	137,438,953,472		53	9,007,199,254,740,992
6	64		22	4,194,304		38	274,877,906,944		54	18,014,398,509,481,984
7	128		23	8,388,608		39	549,755,813,888		55	36,028,797,018,963,968
8	256		24	16,777,216		40	1,099,511,627,776		56	72,057,594,037,927,936
9	512		25	33,554,432		41	2,199,023,255,552		57	144,115,188,075,855,872
10	1,024		26	67,108,864		42	4,398,046,511,104		58	288,230,376,151,711,744
11	2,048		27	134,217,728		43	8,796,093,022,208		59	576,460,752,303,423,488
12	4,096		28	268,435,456		44	17,592,186,044,416		60	1,152,921,504,606,846,976
13	8,192		29	536,870,912		45	35,184,372,088,832		61	2,305,843,009,213,693,952
14	16,384		30	1,073,741,824		46	70,368,744,177,664		62	4,611,686,018,427,387,904
15	32,768		31	2,147,483,648		47	140,737,488,355,328		63	9,223,372,036,854,775,808

Last digits

Starting with 2 the last digit is periodic with period 4, with the cycle 2–4–8–6–, and starting with 4 the last two digits are periodic with period 20. These patterns are generally true of any power, with respect to any base. The pattern continues where each pattern has starting point 2^k, and the period is the multiplicative order of 2 modulo 5^k, which is φ(5^k) = 4 × 5^k−1 (see Multiplicative group of integers modulo n).^{[citation needed]}

Powers of 1024

(sequence A140300 in the OEIS)

The first few powers of 2¹⁰ are slightly larger than those same powers of 1000 (10³). The powers of 2¹⁰ values that have less than 25% deviation are listed below:

2^0	=	1	= 1000^0	(0% deviation)
2^10	=	1 024	≈ 1000^1	(2.4% deviation)
2^20	=	1 048 576	≈ 1000^2	(4.9% deviation)
2^30	=	1 073 741 824	≈ 1000^3	(7.4% deviation)
2^40	=	1 099 511 627 776	≈ 1000^4	(10.0% deviation)
2^50	=	1 125 899 906 842 624	≈ 1000^5	(12.6% deviation)
2^60	=	1 152 921 504 606 846 976	≈ 1000^6	(15.3% deviation)
2^70	=	1 180 591 620 717 411 303 424	≈ 1000^7	(18.1% deviation)
2^80	=	1 208 925 819 614 629 174 706 176	≈ 1000^8	(20.9% deviation)
2^90	=	1 237 940 039 285 380 274 899 124 224	≈ 1000^9	(23.8% deviation)

It takes approximately 17 powers of 1024 to reach 50% deviation and approximately 29 powers of 1024 to reach 100% deviation of the same powers of 1000.^{[citation needed]}

See also: Binary prefixes and IEEE 1541-2002

Powers of two whose exponents are powers of two

Because data (specifically integers) and the addresses of data are stored using the same hardware, and the data is stored in one or more octets (2³), double exponentials of two are common. The first 20 of them are:

n	2^n	2^2n (sequence A001146 in the OEIS)	digits
0	1	2	1
1	2	4	1
2	4	16	2
3	8	256	3
4	16	65,536	5
5	32	4,294,967,296	10
6	64	18,​446,​744,​073,​709,​551,​616	20
7	128	340,​282,​366,​920,​938,​463,​463,​374,​607,​431,​768,​211,​456	39
8	256	115,​792,​089,​237,​316,​195,​423,​570,​9...4,​039,​457,​584,​007,​913,​129,​639,​936	78
9	512	13,​407,​807,​929,​942,​597,​099,​574,​02...1,​946,​569,​946,​433,​649,​006,​084,​096	155
10	1,024	179,​769,​313,​486,​231,​590,​772,​930,​5...6,​304,​835,​356,​329,​624,​224,​137,​216	309
11	2,048	32,​317,​006,​071,​311,​007,​300,​714,​87...8,​193,​555,​853,​611,​059,​596,​230,​656	617
12	4,096	1,​044,​388,​881,​413,​152,​506,​691,​752,​...0,​243,​804,​708,​340,​403,​154,​190,​336	1,234
13	8,192	1,​090,​748,​135,​619,​415,​929,​462,​984,​...1,​997,​186,​505,​665,​475,​715,​792,​896	2,467
14	16,384	1,​189,​731,​495,​357,​231,​765,​085,​759,​...2,​460,​447,​027,​290,​669,​964,​066,​816	4,933
15	32,768	1,​415,​461,​031,​044,​954,​789,​001,​553,​...7,​541,​122,​668,​104,​633,​712,​377,​856	9,865
16	65,536	2,​003,​529,​930,​406,​846,​464,​979,​072,​...2,​339,​445,​587,​895,​905,​719,​156,​736	19,729
17	131,072	4,​014,​132,​182,​036,​063,​039,​166,​060,​...1,​850,​665,​812,​318,​570,​934,​173,​696	39,457
18	262,144	16,​113,​257,​174,​857,​604,​736,​195,​72...0,​753,​862,​605,​349,​934,​298,​300,​416	78,914
19	524,288	259,​637,​056,​783,​100,​077,​612,​659,​6...1,​369,​814,​364,​528,​226,​185,​773,​056	157,827

Also see tetration and lower hyperoperations.

Last digits for powers of two whose exponents are powers of two

All of these numbers end in 6. Starting with 16 the last two digits are periodic with period 4, with the cycle 16–56–36–96–, and starting with 16 the last three digits are periodic with period 20. These patterns are generally true of any power, with respect to any base. The pattern continues where each pattern has starting point 2^k, and the period is the multiplicative order of 2 modulo 5^k, which is φ(5^k) = 4 × 5^k−1 (see Multiplicative group of integers modulo n).^{[citation needed]}

Facts about powers of two whose exponents are powers of two

In a connection with nimbers, these numbers are often called Fermat 2-powers.

The numbers ${\displaystyle 2^{2^{n}}}$  form an irrationality sequence: for every sequence ${\displaystyle x_{i}}$  of positive integers, the series

${\displaystyle \sum _{i=0}^{\infty }{\frac {1}{2^{2^{i}}x_{i}}}={\frac {1}{2x_{0}}}+{\frac {1}{4x_{1}}}+{\frac {1}{16x_{2}}}+\cdots }$ 

converges to an irrational number. Despite the rapid growth of this sequence, it is the slowest-growing irrationality sequence known.^[3]

Powers of two whose exponents are powers of two in computer science

Several of these numbers represent the number of values representable using common computer data types. For example, a 32-bit word consisting of 4 bytes can represent 2³² distinct values, which can either be regarded as mere bit-patterns, or are more commonly interpreted as the unsigned numbers from 0 to 2³² − 1, or as the range of signed numbers between −2³¹ and 2³¹ − 1. For more about representing signed numbers see two's complement.

Selected powers of two

2² = 4

The number that is the square of two. Also the first power of two tetration of two.

2⁸ = 256

The number of values represented by the 8 bits in a byte, more specifically termed as an octet. (The term byte is often defined as a collection of bits rather than the strict definition of an 8-bit quantity, as demonstrated by the term kilobyte.)

2¹⁰ = 1,024

The binary approximation of the kilo-, or 1,000 multiplier, which causes a change of prefix. For example: 1,024 bytes = 1 kilobyte (or kibibyte).

2¹² = 4,096

The hardware page size of an Intel x86-compatible processor.

2¹⁵ = 32,768

The number of non-negative values for a signed 16-bit integer.

2¹⁶ = 65,536

Main article: 65,536

The number of distinct values representable in a single word on a 16-bit processor, such as the original x86 processors.^[4]

The maximum range of a short integer variable in the C#, Java, and SQL programming languages. The maximum range of a Word or Smallint variable in the Pascal programming language.

The number of binary relations on a 4-element set.

2²⁰ = 1,048,576

The binary approximation of the mega-, or 1,000,000 multiplier, which causes a change of prefix. For example: 1,048,576 bytes = 1 megabyte (or mebibyte).

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, where colors are defined by three values (red, green and blue) independently ranging from 0 (00) to 255 (FF) inclusive. This gives 8 bits for each channel, or 24 bits in total; for example, pure black is #000000, pure white is #FFFFFF. The space of all possible colors, 16,777,216, can be determined by 16⁶ (6 digits with 16 possible values for each), 256³ (3 channels with 256 possible values for each), or 2²⁴ (24 bits with 2 possible values for each).

The size of the largest unsigned integer or address in computers with 24-bit registers or data buses.

2²⁹ = 536,870,912

The largest power of two with distinct digits in base ten.^[5]

2³⁰ = 1,073,741,824

The binary approximation of the giga-, or 1,000,000,000 multiplier, which causes a change of prefix. For example, 1,073,741,824 bytes = 1 gigabyte (or gibibyte).

2³¹ = 2,147,483,648

See also: 2,147,483,647

The number of non-negative values for a signed 32-bit integer. Since Unix time is measured in seconds since January 1, 1970, it will run out at 2,147,483,647 seconds or 03:14:07 UTC on Tuesday, 19 January 2038 on 32-bit computers running Unix, a problem known as the year 2038 problem.

2³² = 4,294,967,296

See also: 4,294,967,295

The number of distinct values representable in a single word on a 32-bit processor.^[6] Or, the number of values representable in a doubleword on a 16-bit processor, such as the original x86 processors.^[4]

The range of an int variable in the Java, C#, and SQL programming languages.

The range of a Cardinal or Integer variable in the Pascal programming language.

The minimum range of a long integer variable in the C and C++ programming languages.

The total number of IP addresses under IPv4. Although this is a seemingly large number, the number of available 32-bit IPv4 addresses has been exhausted (but not for IPv6 addresses).

The number of binary operations with domain equal to any 4-element set, such as GF(4).

2⁴⁰ = 1,099,511,627,776

The binary approximation of the tera-, or 1,000,000,000,000 multiplier, which causes a change of prefix. For example, 1,099,511,627,776 bytes = 1 terabyte or tebibyte.

2⁵⁰ = 1,125,899,906,842,624

The binary approximation of the peta-, or 1,000,000,000,000,000 multiplier. 1,125,899,906,842,624 bytes = 1 petabyte or pebibyte.

2⁵³ = 9,007,199,254,740,992

The number until which all integer values can exactly be represented in IEEE double precision floating-point format. Also the first power of 2 to start with the digit 9 in decimal.

2⁵⁶ = 72,057,594,037,927,936

The number of different possible keys in the obsolete 56 bit DES symmetric cipher.

2⁶⁰ = 1,152,921,504,606,846,976

The binary approximation of the exa-, or 1,000,000,000,000,000,000 multiplier. 1,152,921,504,606,846,976 bytes = 1 exabyte or exbibyte.

2⁶³ = 9,223,372,036,854,775,808

The number of non-negative values for a signed 64-bit integer.

2⁶³ − 1, a common maximum value (equivalently the number of positive values) for a signed 64-bit integer in programming languages.

2⁶⁴ = 18,446,744,073,709,551,616

The number of distinct values representable in a single word on a 64-bit processor. Or, the number of values representable in a doubleword on a 32-bit processor. Or, the number of values representable in a quadword on a 16-bit processor, such as the original x86 processors.^[4]

The range of a long variable in the Java and C# programming languages.

The range of a Int64 or QWord variable in the Pascal programming language.

The total number of IPv6 addresses generally given to a single LAN or subnet.

2⁶⁴ − 1, the number of grains of rice on a chessboard, according to the old story, where the first square contains one grain of rice and each succeeding square twice as many as the previous square. For this reason the number is sometimes known as the "chess number".

2⁶⁴ − 1 is also the number of moves required to complete the legendary 64-disk version of the Tower of Hanoi.

2⁶⁸ = 295,147,905,179,352,825,856

The first power of 2 to contain all decimal digits. (sequence A137214 in the OEIS)

2⁷⁰ = 1,180,591,620,717,411,303,424

The binary approximation of the zetta-, or 1,000,000,000,000,000,000,000 multiplier. 1,180,591,620,717,411,303,424 bytes = 1 zettabyte (or zebibyte).

2⁸⁰ = 1,208,925,819,614,629,174,706,176

The binary approximation of the yotta-, or 1,000,000,000,000,000,000,000,000 multiplier. 1,208,925,819,614,629,174,706,176 bytes = 1 yottabyte (or yobibyte).

2⁸⁶ = 77,371,252,455,336,267,181,195,264

2⁸⁶ is conjectured to be the largest power of two not containing a zero in decimal.^[7]

2⁹⁶ = 79,228,162,514,264,337,593,543,950,336

The total number of IPv6 addresses generally given to a local Internet registry. In CIDR notation, ISPs are given a /32, which means that 128-32=96 bits are available for addresses (as opposed to network designation). Thus, 2⁹⁶ addresses.

2¹⁰⁸ = 324,​518,​553,​658,​426,​726,​783,​156,​020,​576,​256

The largest known power of 2 not containing a 9 in decimal. (sequence A035064 in the OEIS)

2¹²⁶ = 85,​070,​591,​730,​234,​615,​865,​843,​651,​857,​942,​052,​864

The largest known power of 2 not containing a pair of consecutive equal digits. (sequence A050723 in the OEIS)

2¹²⁸ = 340,282,366,920,938,463,463,374,607,431,768,211,456

The total number of IP addresses available under IPv6. Also the number of distinct universally unique identifiers (UUIDs).

2¹⁶⁸ = 374,144,419,156,711,147,060,143,317,175,368,453,031,918,731,001,856

The largest known power of 2 not containing all decimal digits (the digit 2 is missing in this case). (sequence A137214 in the OEIS)

2¹⁹² = 6,277,101,735,386,680,763,835,789,423,207,666,416,102,355,444,464,034,512,896

The total number of different possible keys in the AES 192-bit key space (symmetric cipher).

2²²⁹ = 862,718,293,348,820,473,429,344,482,784,628,181,556,388,621,521,298,319,395,315,527,974,912

2²²⁹ is the largest known power of two containing the least number of zeros relative to its power. It is conjectured by Metin Sariyar that every digit 0 to 9 is inclined to appear an equal number of times in the decimal expansion of power of two as the power increases. (sequence A330024 in the OEIS)

2²⁵⁶ = 115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,639,936

The total number of different possible keys in the AES 256-bit key space (symmetric cipher).

2^1,024 = 179,769,313,486,231,590,772,930,...,304,835,356,329,624,224,137,216

The maximum number that can fit in a 64-bit IEEE double-precision floating-point format (approximately 1.797×10³⁰⁸), and hence the maximum number that can be represented by many programs, for example Microsoft Excel.

2^16,384 = 1,189,731,495,357,231,765,085,75...,460,447,027,290,669,964,066,816

The maximum number that can fit in a 128-bit IEEE quadruple-precision floating-point format (approximately 1.189×10⁴⁹³²).

2^262,144 = 16,113,257,174,857,604,736,195,7...,753,862,605,349,934,298,300,416

The maximum number that can fit in a 256-bit IEEE octuple-precision floating-point format (approximately 1.611×10⁷⁸⁹¹³).

2^82,589,933 = 1,488,944,457,420,413,255,478,06...,074,037,951,210,325,217,902,592

One more than the largest known prime number as of June 2023^[update]. It has 24,862,048 digits.^[8]

Powers of two in music theory

In musical notation, all unmodified note values have a duration equal to a whole note divided by a power of two; for example a half note (1/2), a quarter note (1/4), an eighth note (1/8) and a sixteenth note (1/16). Dotted or otherwise modified notes have other durations. In time signatures the lower numeral, the beat unit, which can be seen as the denominator of a fraction, is almost always a power of two.

If the ratio of frequencies of two pitches is a power of two, then the interval between those pitches is full octaves. In this case, the corresponding notes have the same name.

Other properties

 

As each increase in dimension doubles the number of shapes, the sum of coefficients on each row of Pascal's triangle is a power of two

 

The sum of powers of two from zero to a given power, inclusive, is 1 less than the next power of two, whereas the sum of powers of two from minus-infinity to a given power, inclusive, equals the next power of two

The sum of all n-choose binomial coefficients is equal to 2ⁿ. Consider the set of all n-digit binary integers. Its cardinality is 2ⁿ. It is also the sums of the cardinalities of certain subsets: the subset of integers with no 1s (consisting of a single number, written as n 0s), the subset with a single 1, the subset with two 1s, and so on up to the subset with n 1s (consisting of the number written as n 1s). Each of these is in turn equal to the binomial coefficient indexed by n and the number of 1s being considered (for example, there are 10-choose-3 binary numbers with ten digits that include exactly three 1s).

Currently, powers of two are the only known almost perfect numbers.

The number of vertices of an n-dimensional hypercube is 2ⁿ. Similarly, the number of (n − 1)-faces of an n-dimensional cross-polytope is also 2ⁿ and the formula for the number of x-faces an n-dimensional cross-polytope has is ${\displaystyle 2^{x}{\tbinom {n}{x}}.}$ 

The sum of the reciprocals of the powers of two is 1. The sum of the reciprocals of the squared powers of two (powers of four) is 1/3.

The smallest natural power of two whose decimal representation begins with 7 is^[9]

${\displaystyle 2^{46}=70\ 368\ 744\ 177\ 664.}$ 

Every power of 2 (excluding 1) can be written as the sum of four square numbers in 24 ways. The powers of 2 are the natural numbers greater than 1 that can be written as the sum of four square numbers in the fewest ways.

As a real polynomial, aⁿ + bⁿ is irreducible, if and only if n is a power of two. (If n is odd, then aⁿ + bⁿ is divisible by a+n, and if n is even but not a power of 2, then n can be written as n=mp, where m is odd, and thus ${\displaystyle a^{n}+b^{n}=(a^{p})^{m}+(b^{p})^{m}}$ , which is divisible by a^p + b^p.) But in the domain of complex numbers, the polynomial ${\displaystyle a^{2n}+b^{2n}}$  (where n>=1) can always be factorized as ${\displaystyle a^{2n}+b^{2n}=(a^{n}+b^{n}i)\cdot (a^{n}-b^{n}i)}$ , even if n is a power of two.

See also

• Fermi–Dirac prime
• Gould's sequence
• Binary logarithm
• Power of three
• Power of 10

References

1. Lipschutz, Seymour (1982). Schaum's Outline of Theory and Problems of Essential Computer Mathematics. New York: McGraw-Hill. p. 3. ISBN 0-07-037990-4.
2. Sewell, Michael J. (1997). Mathematics Masterclasses. Oxford: Oxford University Press. p. 78. ISBN 0-19-851494-8.
3. Guy, Richard K. (2004), "E24 Irrationality sequences", Unsolved problems in number theory (3rd ed.), Springer-Verlag, p. 346, ISBN 0-387-20860-7, Zbl 1058.11001, from the original on 2016-04-28
4. Though they vary in word size, all x86 processors use the term "word" to mean 16 bits; thus, a 32-bit x86 processor refers to its native wordsize as a dword
5. Prime Curios!: 536870912 "Prime Curios! 536870912". from the original on 2017-09-05. Retrieved 2017-09-05.
6. . www.vaughns-1-pagers.com. Archived from the original on August 12, 2015.
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8. "Mersenne Prime Discovery - 2^82589933-1 is Prime!". www.mersenne.org.
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